958. Check Completeness of a Binary Tree - Explanation

Description

You are given the root of a binary tree, determine if it is a complete binary tree.

In a complete binary tree, every level, except possibly the last, is completely filled, and all nodes in the last level are as far left as possible. It can have between 1 and (2^h)^ nodes inclusive at the last level h`.

Example 1:

Input: root = [1,2,3,4,5,6]

Output: true

Example 2:

Input: root = [1,2,3,4,5,null,7]

Output: false

Example 3:

Input: root = [1,2,3,4]

Output: true

Constraints:

1 <= The number of nodes in the tree <= 100.
1 <= Node.val <= 1000

Topics

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Prerequisites

Before attempting this problem, you should be comfortable with:

Binary Trees - Understanding tree structure, nodes, and null children
Breadth-First Search (BFS) - Level-order traversal using a queue to process nodes
Depth-First Search (DFS) - Recursive tree traversal and tracking node indices
Queue Data Structure - FIFO operations for level-by-level tree processing

1. Breadth First Search

Intuition

In a complete binary tree, all nodes are as far left as possible, with no gaps. If we traverse level by level and encounter a null node, all subsequent nodes in the traversal must also be null. If we find any non-null node after a null, the tree is not complete.

Algorithm

Initialize a queue with the root node.
Process nodes in BFS order:
- Dequeue a node and add its left and right children (even if null) to the queue.
- If the dequeued node is null, drain the remaining queue.
- If any non-null node appears after a null, return false.
Return true if the entire queue is processed without finding a non-null after null.

# Definition for a binary tree node.
# class TreeNode:
#     def __init__(self, val=0, left=None, right=None):
#         self.val = val
#         self.left = left
#         self.right = right
class Solution:
    def isCompleteTree(self, root: Optional[TreeNode]) -> bool:
        q = deque([root])
        while q:
            node = q.popleft()
            if node:
                q.append(node.left)
                q.append(node.right)
            else:
                while q:
                    if q.popleft():
                        return False
        return True

/**
 * Definition for a binary tree node.
 * public class TreeNode {
 *     int val;
 *     TreeNode left;
 *     TreeNode right;
 *     TreeNode() {}
 *     TreeNode(int val) { this.val = val; }
 *     TreeNode(int val, TreeNode left, TreeNode right) {
 *         this.val = val;
 *         this.left = left;
 *         this.right = right;
 *     }
 * }
 */
public class Solution {
    public boolean isCompleteTree(TreeNode root) {
        Queue<TreeNode> q = new LinkedList<>();
        q.add(root);

        while (!q.isEmpty()) {
            TreeNode node = q.poll();
            if (node != null) {
                q.add(node.left);
                q.add(node.right);
            } else {
                while (!q.isEmpty()) {
                    if (q.poll() != null) {
                        return false;
                    }
                }
            }
        }
        return true;
    }
}

/**
 * Definition for a binary tree node.
 * struct TreeNode {
 *     int val;
 *     TreeNode *left;
 *     TreeNode *right;
 *     TreeNode() : val(0), left(nullptr), right(nullptr) {}
 *     TreeNode(int x) : val(x), left(nullptr), right(nullptr) {}
 *     TreeNode(int x, TreeNode *left, TreeNode *right) : val(x), left(left), right(right) {}
 * };
 */
class Solution {
public:
    bool isCompleteTree(TreeNode* root) {
        queue<TreeNode*> q;
        q.push(root);

        while (!q.empty()) {
            TreeNode* node = q.front();
            q.pop();
            if (node) {
                q.push(node->left);
                q.push(node->right);
            } else {
                while (!q.empty()) {
                    if (q.front()) {
                        return false;
                    }
                    q.pop();
                }
            }
        }
        return true;
    }
};

/**
 * Definition for a binary tree node.
 * class TreeNode {
 *     constructor(val = 0, left = null, right = null) {
 *         this.val = val;
 *         this.left = left;
 *         this.right = right;
 *     }
 * }
 */
class Solution {
    /**
     * @param {TreeNode} root
     * @return {boolean}
     */
    isCompleteTree(root) {
        const queue = new Queue([root]);

        while (!queue.isEmpty()) {
            const node = queue.pop();
            if (node) {
                queue.push(node.left);
                queue.push(node.right);
            } else {
                while (!queue.isEmpty()) {
                    if (queue.pop()) {
                        return false;
                    }
                }
            }
        }
        return true;
    }
}

/**
 * Definition for a binary tree node.
 * public class TreeNode {
 *     public int val;
 *     public TreeNode left;
 *     public TreeNode right;
 *     public TreeNode(int val=0, TreeNode left=null, TreeNode right=null) {
 *         this.val = val;
 *         this.left = left;
 *         this.right = right;
 *     }
 * }
 */
public class Solution {
    public bool IsCompleteTree(TreeNode root) {
        Queue<TreeNode> q = new Queue<TreeNode>();
        q.Enqueue(root);

        while (q.Count > 0) {
            TreeNode node = q.Dequeue();
            if (node != null) {
                q.Enqueue(node.left);
                q.Enqueue(node.right);
            } else {
                while (q.Count > 0) {
                    if (q.Dequeue() != null) {
                        return false;
                    }
                }
            }
        }

        return true;
    }
}

/**
 * Definition for a binary tree node.
 * type TreeNode struct {
 *     Val int
 *     Left *TreeNode
 *     Right *TreeNode
 * }
 */
func isCompleteTree(root *TreeNode) bool {
    q := []*TreeNode{root}

    for len(q) > 0 {
        node := q[0]
        q = q[1:]
        if node != nil {
            q = append(q, node.Left)
            q = append(q, node.Right)
        } else {
            for len(q) > 0 {
                if q[0] != nil {
                    return false
                }
                q = q[1:]
            }
        }
    }
    return true
}

/**
 * Example:
 * var ti = TreeNode(5)
 * var v = ti.`val`
 * Definition for a binary tree node.
 * class TreeNode(var `val`: Int) {
 *     var left: TreeNode? = null
 *     var right: TreeNode? = null
 * }
 */
class Solution {
    fun isCompleteTree(root: TreeNode?): Boolean {
        val q = ArrayDeque<TreeNode?>()
        q.add(root)

        while (q.isNotEmpty()) {
            val node = q.removeFirst()
            if (node != null) {
                q.add(node.left)
                q.add(node.right)
            } else {
                while (q.isNotEmpty()) {
                    if (q.removeFirst() != null) {
                        return false
                    }
                }
            }
        }
        return true
    }
}

/**
 * Definition for a binary tree node.
 * public class TreeNode {
 *     public var val: Int
 *     public var left: TreeNode?
 *     public var right: TreeNode?
 *     public init() { self.val = 0; self.left = nil; self.right = nil; }
 *     public init(_ val: Int) { self.val = val; self.left = nil; self.right = nil; }
 *     public init(_ val: Int, _ left: TreeNode?, _ right: TreeNode?) {
 *         self.val = val
 *         self.left = left
 *         self.right = right
 *     }
 * }
 */
class Solution {
    func isCompleteTree(_ root: TreeNode?) -> Bool {
        var queue = [root]

        while !queue.isEmpty {
            let node = queue.removeFirst()
            if let node = node {
                queue.append(node.left)
                queue.append(node.right)
            } else {
                while !queue.isEmpty {
                    if queue.removeFirst() != nil {
                        return false
                    }
                }
            }
        }
        return true
    }
}

Time & Space Complexity

Time complexity: $O(n)$
Space complexity: $O(n)$

2. Breadth First Search (Optimal)

Intuition

This is a cleaner version of the BFS approach. Instead of draining the queue after seeing null, we use a flag to track whether a null has been seen. If we encounter a non-null node after the flag is set, the tree is incomplete.

Algorithm

Initialize a queue with the root and a boolean flag nullSeen = false.
Process each node from the queue:
- If the node is non-null and nullSeen is true, return false.
- If the node is non-null, add both children to the queue.
- If the node is null, set nullSeen = true.
Return true after processing all nodes.

# Definition for a binary tree node.
# class TreeNode:
#     def __init__(self, val=0, left=None, right=None):
#         self.val = val
#         self.left = left
#         self.right = right
class Solution:
    def isCompleteTree(self, root: Optional[TreeNode]) -> bool:
        q = deque([root])
        nullSeen = False
        while q:
            node = q.popleft()
            if node:
                if nullSeen:
                    return False
                q.append(node.left)
                q.append(node.right)
            else:
                nullSeen = True
        return True

/**
 * Definition for a binary tree node.
 * public class TreeNode {
 *     int val;
 *     TreeNode left;
 *     TreeNode right;
 *     TreeNode() {}
 *     TreeNode(int val) { this.val = val; }
 *     TreeNode(int val, TreeNode left, TreeNode right) {
 *         this.val = val;
 *         this.left = left;
 *         this.right = right;
 *     }
 * }
 */
public class Solution {
    public boolean isCompleteTree(TreeNode root) {
        Queue<TreeNode> q = new LinkedList<>();
        q.add(root);
        boolean nullSeen = false;

        while (!q.isEmpty()) {
            TreeNode node = q.poll();
            if (node != null) {
                if (nullSeen) return false;
                q.add(node.left);
                q.add(node.right);
            } else {
                nullSeen = true;
            }
        }
        return true;
    }
}

/**
 * Definition for a binary tree node.
 * struct TreeNode {
 *     int val;
 *     TreeNode *left;
 *     TreeNode *right;
 *     TreeNode() : val(0), left(nullptr), right(nullptr) {}
 *     TreeNode(int x) : val(x), left(nullptr), right(nullptr) {}
 *     TreeNode(int x, TreeNode *left, TreeNode *right) : val(x), left(left), right(right) {}
 * };
 */
class Solution {
public:
    bool isCompleteTree(TreeNode* root) {
        queue<TreeNode*> q;
        q.push(root);
        bool nullSeen = false;

        while (!q.empty()) {
            TreeNode* node = q.front();
            q.pop();
            if (node) {
                if (nullSeen) return false;
                q.push(node->left);
                q.push(node->right);
            } else {
                nullSeen = true;
            }
        }
        return true;
    }
};

/**
 * Definition for a binary tree node.
 * class TreeNode {
 *     constructor(val = 0, left = null, right = null) {
 *         this.val = val;
 *         this.left = left;
 *         this.right = right;
 *     }
 * }
 */
class Solution {
    /**
     * @param {TreeNode} root
     * @return {boolean}
     */
    isCompleteTree(root) {
        const queue = new Queue([root]);
        let nullSeen = false;

        while (!queue.isEmpty()) {
            const node = queue.pop();
            if (node) {
                if (nullSeen) return false;
                queue.push(node.left);
                queue.push(node.right);
            } else {
                nullSeen = true;
            }
        }
        return true;
    }
}

/**
 * Definition for a binary tree node.
 * public class TreeNode {
 *     public int val;
 *     public TreeNode left;
 *     public TreeNode right;
 *     public TreeNode(int val=0, TreeNode left=null, TreeNode right=null) {
 *         this.val = val;
 *         this.left = left;
 *         this.right = right;
 *     }
 * }
 */
public class Solution {
    public bool IsCompleteTree(TreeNode root) {
        Queue<TreeNode> q = new Queue<TreeNode>();
        q.Enqueue(root);
        bool nullSeen = false;

        while (q.Count > 0) {
            TreeNode node = q.Dequeue();
            if (node != null) {
                if (nullSeen) {
                    return false;
                }
                q.Enqueue(node.left);
                q.Enqueue(node.right);
            } else {
                nullSeen = true;
            }
        }

        return true;
    }
}

/**
 * Definition for a binary tree node.
 * type TreeNode struct {
 *     Val int
 *     Left *TreeNode
 *     Right *TreeNode
 * }
 */
func isCompleteTree(root *TreeNode) bool {
    q := []*TreeNode{root}
    nullSeen := false

    for len(q) > 0 {
        node := q[0]
        q = q[1:]
        if node != nil {
            if nullSeen {
                return false
            }
            q = append(q, node.Left)
            q = append(q, node.Right)
        } else {
            nullSeen = true
        }
    }
    return true
}

/**
 * Example:
 * var ti = TreeNode(5)
 * var v = ti.`val`
 * Definition for a binary tree node.
 * class TreeNode(var `val`: Int) {
 *     var left: TreeNode? = null
 *     var right: TreeNode? = null
 * }
 */
class Solution {
    fun isCompleteTree(root: TreeNode?): Boolean {
        val q = ArrayDeque<TreeNode?>()
        q.add(root)
        var nullSeen = false

        while (q.isNotEmpty()) {
            val node = q.removeFirst()
            if (node != null) {
                if (nullSeen) return false
                q.add(node.left)
                q.add(node.right)
            } else {
                nullSeen = true
            }
        }
        return true
    }
}

/**
 * Definition for a binary tree node.
 * public class TreeNode {
 *     public var val: Int
 *     public var left: TreeNode?
 *     public var right: TreeNode?
 *     public init() { self.val = 0; self.left = nil; self.right = nil; }
 *     public init(_ val: Int) { self.val = val; self.left = nil; self.right = nil; }
 *     public init(_ val: Int, _ left: TreeNode?, _ right: TreeNode?) {
 *         self.val = val
 *         self.left = left
 *         self.right = right
 *     }
 * }
 */
class Solution {
    func isCompleteTree(_ root: TreeNode?) -> Bool {
        var queue: [TreeNode?] = [root]
        var nullSeen = false

        while !queue.isEmpty {
            let node = queue.removeFirst()
            if let node = node {
                if nullSeen { return false }
                queue.append(node.left)
                queue.append(node.right)
            } else {
                nullSeen = true
            }
        }
        return true
    }
}

Time & Space Complexity

Time complexity: $O(n)$
Space complexity: $O(n)$

3. Depth First Search (Two Pass)

Intuition

A complete binary tree with n nodes has a specific property: if we number nodes starting from 0 (root) where a node at index i has children at 2i+1 and 2i+2, then every node's index must be less than n. First count all nodes, then verify that no node has an index >= n.

Algorithm

First pass: Count the total number of nodes in the tree using DFS.
Second pass: Perform DFS with indices, starting from (root, 0).
- If a node is null, return true.
- If a node's index >= total count, return false (gap exists).
- Recursively check left child (index 2i+1) and right child (index 2i+2).
Return true if all nodes pass the index check.

# Definition for a binary tree node.
# class TreeNode:
#     def __init__(self, val=0, left=None, right=None):
#         self.val = val
#         self.left = left
#         self.right = right
class Solution:
    def isCompleteTree(self, root: Optional[TreeNode]) -> bool:
        def dfs(node, index, n):
            if not node:
                return True
            if index >= n:
                return False

            left = dfs(node.left, 2 * index + 1, n)
            right = dfs(node.right, 2 * index + 2, n)
            return left and right

        def countNodes(node):
            if not node:
                return 0
            return 1 + countNodes(node.left) + countNodes(node.right)

        n = countNodes(root)
        return dfs(root, 0, n)

/**
 * Definition for a binary tree node.
 * public class TreeNode {
 *     int val;
 *     TreeNode left;
 *     TreeNode right;
 *     TreeNode() {}
 *     TreeNode(int val) { this.val = val; }
 *     TreeNode(int val, TreeNode left, TreeNode right) {
 *         this.val = val;
 *         this.left = left;
 *         this.right = right;
 *     }
 * }
 */
public class Solution {
    private int countNodes(TreeNode root) {
        if (root == null) return 0;
        return 1 + countNodes(root.left) + countNodes(root.right);
    }

    private boolean dfs(TreeNode node, int index, int n) {
        if (node == null) return true;
        if (index >= n) return false;
        return dfs(node.left, 2 * index + 1, n) && dfs(node.right, 2 * index + 2, n);
    }

    public boolean isCompleteTree(TreeNode root) {
        int n = countNodes(root);
        return dfs(root, 0, n);
    }
}

/**
 * Definition for a binary tree node.
 * struct TreeNode {
 *     int val;
 *     TreeNode *left;
 *     TreeNode *right;
 *     TreeNode() : val(0), left(nullptr), right(nullptr) {}
 *     TreeNode(int x) : val(x), left(nullptr), right(nullptr) {}
 *     TreeNode(int x, TreeNode *left, TreeNode *right) : val(x), left(left), right(right) {}
 * };
 */
class Solution {
public:
    int countNodes(TreeNode* root) {
        if (!root) return 0;
        return 1 + countNodes(root->left) + countNodes(root->right);
    }

    bool dfs(TreeNode* node, int index, int n) {
        if (!node) return true;
        if (index >= n) return false;
        return dfs(node->left, 2 * index + 1, n) && dfs(node->right, 2 * index + 2, n);
    }

    bool isCompleteTree(TreeNode* root) {
        int n = countNodes(root);
        return dfs(root, 0, n);
    }
};

/**
 * Definition for a binary tree node.
 * class TreeNode {
 *     constructor(val = 0, left = null, right = null) {
 *         this.val = val;
 *         this.left = left;
 *         this.right = right;
 *     }
 * }
 */
class Solution {
    /**
     * @param {TreeNode} root
     * @return {boolean}
     */
    isCompleteTree(root) {
        const countNodes = (node) => {
            if (!node) return 0;
            return 1 + countNodes(node.left) + countNodes(node.right);
        };

        const dfs = (node, index, n) => {
            if (!node) return true;
            if (index >= n) return false;
            return (
                dfs(node.left, 2 * index + 1, n) &&
                dfs(node.right, 2 * index + 2, n)
            );
        };

        const n = countNodes(root);
        return dfs(root, 0, n);
    }
}

/**
 * Definition for a binary tree node.
 * public class TreeNode {
 *     public int val;
 *     public TreeNode left;
 *     public TreeNode right;
 *     public TreeNode(int val=0, TreeNode left=null, TreeNode right=null) {
 *         this.val = val;
 *         this.left = left;
 *         this.right = right;
 *     }
 * }
 */
public class Solution {
    public bool IsCompleteTree(TreeNode root) {
        int n = CountNodes(root);
        return Dfs(root, 0, n);
    }

    private bool Dfs(TreeNode node, int index, int n) {
        if (node == null) return true;
        if (index >= n) return false;
        return Dfs(node.left, 2 * index + 1, n) && Dfs(node.right, 2 * index + 2, n);
    }

    private int CountNodes(TreeNode node) {
        if (node == null) return 0;
        return 1 + CountNodes(node.left) + CountNodes(node.right);
    }
}

/**
 * Definition for a binary tree node.
 * type TreeNode struct {
 *     Val int
 *     Left *TreeNode
 *     Right *TreeNode
 * }
 */
func isCompleteTree(root *TreeNode) bool {
    var countNodes func(node *TreeNode) int
    countNodes = func(node *TreeNode) int {
        if node == nil {
            return 0
        }
        return 1 + countNodes(node.Left) + countNodes(node.Right)
    }

    var dfs func(node *TreeNode, index, n int) bool
    dfs = func(node *TreeNode, index, n int) bool {
        if node == nil {
            return true
        }
        if index >= n {
            return false
        }
        return dfs(node.Left, 2*index+1, n) && dfs(node.Right, 2*index+2, n)
    }

    n := countNodes(root)
    return dfs(root, 0, n)
}

/**
 * Example:
 * var ti = TreeNode(5)
 * var v = ti.`val`
 * Definition for a binary tree node.
 * class TreeNode(var `val`: Int) {
 *     var left: TreeNode? = null
 *     var right: TreeNode? = null
 * }
 */
class Solution {
    fun isCompleteTree(root: TreeNode?): Boolean {
        fun countNodes(node: TreeNode?): Int {
            if (node == null) return 0
            return 1 + countNodes(node.left) + countNodes(node.right)
        }

        fun dfs(node: TreeNode?, index: Int, n: Int): Boolean {
            if (node == null) return true
            if (index >= n) return false
            return dfs(node.left, 2 * index + 1, n) && dfs(node.right, 2 * index + 2, n)
        }

        val n = countNodes(root)
        return dfs(root, 0, n)
    }
}

/**
 * Definition for a binary tree node.
 * public class TreeNode {
 *     public var val: Int
 *     public var left: TreeNode?
 *     public var right: TreeNode?
 *     public init() { self.val = 0; self.left = nil; self.right = nil; }
 *     public init(_ val: Int) { self.val = val; self.left = nil; self.right = nil; }
 *     public init(_ val: Int, _ left: TreeNode?, _ right: TreeNode?) {
 *         self.val = val
 *         self.left = left
 *         self.right = right
 *     }
 * }
 */
class Solution {
    func isCompleteTree(_ root: TreeNode?) -> Bool {
        func countNodes(_ node: TreeNode?) -> Int {
            guard let node = node else { return 0 }
            return 1 + countNodes(node.left) + countNodes(node.right)
        }

        func dfs(_ node: TreeNode?, _ index: Int, _ n: Int) -> Bool {
            guard let node = node else { return true }
            if index >= n { return false }
            return dfs(node.left, 2 * index + 1, n) && dfs(node.right, 2 * index + 2, n)
        }

        let n = countNodes(root)
        return dfs(root, 0, n)
    }
}

Time & Space Complexity

Time complexity: $O(n)$
Space complexity: $O(n)$ for recursion stack.

4. Depth First Search (Optimal)

Intuition

We can verify completeness in a single DFS pass by tracking the tree's height and whether we have seen a "short" path (one that ends before the maximum depth). In a complete tree, all paths to null nodes at the deepest level must appear before any paths ending one level higher, when traversing left to right.

Algorithm

Track the tree height (initialized to 0) and a flag nullSeen (for detecting gaps).
Perform DFS, tracking the current height:
- When reaching a null node, set treeHgt if not set.
- If the null is at height (treeHgt - 1), mark nullSeen = true.
- If the null is at treeHgt after nullSeen is true, return false (gap detected).
Recursively check left then right children.
Return true if no gaps are detected.

# Definition for a binary tree node.
# class TreeNode:
#     def __init__(self, val=0, left=None, right=None):
#         self.val = val
#         self.left = left
#         self.right = right
class Solution:
    def isCompleteTree(self, root: Optional[TreeNode]) -> bool:
        treeHgt = 0
        nullSeen = False

        def dfs(node, hgt):
            nonlocal treeHgt, nullSeen
            if not node:
                if treeHgt == 0:
                    treeHgt = hgt
                elif hgt == treeHgt - 1:
                    nullSeen = True
                elif hgt != treeHgt:
                    return False
                return not (hgt == treeHgt and nullSeen)

            return dfs(node.left, hgt + 1) and dfs(node.right, hgt + 1)

        return dfs(root, 0)

/**
 * Definition for a binary tree node.
 * public class TreeNode {
 *     int val;
 *     TreeNode left;
 *     TreeNode right;
 *     TreeNode() {}
 *     TreeNode(int val) { this.val = val; }
 *     TreeNode(int val, TreeNode left, TreeNode right) {
 *         this.val = val;
 *         this.left = left;
 *         this.right = right;
 *     }
 * }
 */
public class Solution {
    private int treeHgt = 0;
    private boolean nullSeen = false;

    private boolean dfs(TreeNode node, int hgt) {
        if (node == null) {
            if (treeHgt == 0) {
                treeHgt = hgt;
            } else if (hgt == treeHgt - 1) {
                nullSeen = true;
            } else if (hgt != treeHgt) {
                return false;
            }
            return !(hgt == treeHgt && nullSeen);
        }

        return dfs(node.left, hgt + 1) && dfs(node.right, hgt + 1);
    }

    public boolean isCompleteTree(TreeNode root) {
        return dfs(root, 0);
    }
}

/**
 * Definition for a binary tree node.
 * struct TreeNode {
 *     int val;
 *     TreeNode *left;
 *     TreeNode *right;
 *     TreeNode() : val(0), left(nullptr), right(nullptr) {}
 *     TreeNode(int x) : val(x), left(nullptr), right(nullptr) {}
 *     TreeNode(int x, TreeNode *left, TreeNode *right) : val(x), left(left), right(right) {}
 * };
 */
class Solution {
public:
    int treeHgt = 0;
    bool nullSeen = false;

    bool dfs(TreeNode* node, int hgt) {
        if (!node) {
            if (treeHgt == 0) {
                treeHgt = hgt;
            } else if (hgt == treeHgt - 1) {
                nullSeen = true;
            } else if (hgt != treeHgt) {
                return false;
            }
            return !(hgt == treeHgt && nullSeen);
        }

        return dfs(node->left, hgt + 1) && dfs(node->right, hgt + 1);
    }

    bool isCompleteTree(TreeNode* root) {
        return dfs(root, 0);
    }
};

/**
 * Definition for a binary tree node.
 * class TreeNode {
 *     constructor(val = 0, left = null, right = null) {
 *         this.val = val;
 *         this.left = left;
 *         this.right = right;
 *     }
 * }
 */
class Solution {
    /**
     * @param {TreeNode} root
     * @return {boolean}
     */
    isCompleteTree(root) {
        let treeHgt = 0;
        let nullSeen = false;

        const dfs = (node, hgt) => {
            if (!node) {
                if (treeHgt === 0) {
                    treeHgt = hgt;
                } else if (hgt === treeHgt - 1) {
                    nullSeen = true;
                } else if (hgt !== treeHgt) {
                    return false;
                }
                return !(hgt === treeHgt && nullSeen);
            }

            return dfs(node.left, hgt + 1) && dfs(node.right, hgt + 1);
        };

        return dfs(root, 0);
    }
}

/**
 * Definition for a binary tree node.
 * public class TreeNode {
 *     public int val;
 *     public TreeNode left;
 *     public TreeNode right;
 *     public TreeNode(int val=0, TreeNode left=null, TreeNode right=null) {
 *         this.val = val;
 *         this.left = left;
 *         this.right = right;
 *     }
 * }
 */
public class Solution {
    private int treeHgt = 0;
    private bool nullSeen = false;

    public bool IsCompleteTree(TreeNode root) {
        return Dfs(root, 0);
    }

    private bool Dfs(TreeNode node, int hgt) {
        if (node == null) {
            if (treeHgt == 0) {
                treeHgt = hgt;
            } else if (hgt == treeHgt - 1) {
                nullSeen = true;
            } else if (hgt != treeHgt) {
                return false;
            }
            return !(hgt == treeHgt && nullSeen);
        }
        return Dfs(node.left, hgt + 1) && Dfs(node.right, hgt + 1);
    }
}

/**
 * Definition for a binary tree node.
 * type TreeNode struct {
 *     Val int
 *     Left *TreeNode
 *     Right *TreeNode
 * }
 */
func isCompleteTree(root *TreeNode) bool {
    treeHgt := 0
    nullSeen := false

    var dfs func(node *TreeNode, hgt int) bool
    dfs = func(node *TreeNode, hgt int) bool {
        if node == nil {
            if treeHgt == 0 {
                treeHgt = hgt
            } else if hgt == treeHgt-1 {
                nullSeen = true
            } else if hgt != treeHgt {
                return false
            }
            return !(hgt == treeHgt && nullSeen)
        }
        return dfs(node.Left, hgt+1) && dfs(node.Right, hgt+1)
    }

    return dfs(root, 0)
}

/**
 * Example:
 * var ti = TreeNode(5)
 * var v = ti.`val`
 * Definition for a binary tree node.
 * class TreeNode(var `val`: Int) {
 *     var left: TreeNode? = null
 *     var right: TreeNode? = null
 * }
 */
class Solution {
    private var treeHgt = 0
    private var nullSeen = false

    fun isCompleteTree(root: TreeNode?): Boolean {
        treeHgt = 0
        nullSeen = false
        return dfs(root, 0)
    }

    private fun dfs(node: TreeNode?, hgt: Int): Boolean {
        if (node == null) {
            if (treeHgt == 0) {
                treeHgt = hgt
            } else if (hgt == treeHgt - 1) {
                nullSeen = true
            } else if (hgt != treeHgt) {
                return false
            }
            return !(hgt == treeHgt && nullSeen)
        }
        return dfs(node.left, hgt + 1) && dfs(node.right, hgt + 1)
    }
}

/**
 * Definition for a binary tree node.
 * public class TreeNode {
 *     public var val: Int
 *     public var left: TreeNode?
 *     public var right: TreeNode?
 *     public init() { self.val = 0; self.left = nil; self.right = nil; }
 *     public init(_ val: Int) { self.val = val; self.left = nil; self.right = nil; }
 *     public init(_ val: Int, _ left: TreeNode?, _ right: TreeNode?) {
 *         self.val = val
 *         self.left = left
 *         self.right = right
 *     }
 * }
 */
class Solution {
    func isCompleteTree(_ root: TreeNode?) -> Bool {
        var treeHgt = 0
        var nullSeen = false

        func dfs(_ node: TreeNode?, _ hgt: Int) -> Bool {
            if node == nil {
                if treeHgt == 0 {
                    treeHgt = hgt
                } else if hgt == treeHgt - 1 {
                    nullSeen = true
                } else if hgt != treeHgt {
                    return false
                }
                return !(hgt == treeHgt && nullSeen)
            }
            return dfs(node?.left, hgt + 1) && dfs(node?.right, hgt + 1)
        }

        return dfs(root, 0)
    }
}

Time & Space Complexity

Time complexity: $O(n)$
Space complexity: $O(n)$ for recursion stack.

Common Pitfalls

Not Adding Null Children to Queue

When using BFS, both left and right children must be added to the queue even if they are null. Skipping null children prevents detection of gaps in the tree, since the null marker is needed to identify when a non-null node appears after a gap.

# Wrong: only adds non-null children
if node.left:
    q.append(node.left)
# Correct: always add both children
q.append(node.left)
q.append(node.right)

Checking Only Immediate Children for Completeness

A tree can have a missing left child but present right child, which violates completeness. Simply checking if both children exist misses cases where a null appears in the middle of a level.

# Wrong: only checks current node's children
if node.left is None and node.right is not None:
    return False
# Correct: track if any null seen, then check for non-null after
if node and nullSeen:
    return False

Wrong Index Formula in DFS Approach

When using array-based indexing (root at 0, children at 2i+1 and 2i+2), using incorrect formulas causes false positives. A common mistake is using 2i and 2i+1, which corresponds to 1-indexed arrays instead of 0-indexed.

# Wrong: 1-indexed formula with 0-indexed array
left_child = 2 * index
right_child = 2 * index + 1
# Correct: 0-indexed formula
left_child = 2 * index + 1
right_child = 2 * index + 2