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Burning Tree

Hard Trees & BST Child + Ancestor Handling Original on GeeksforGeeks

Burning Tree is a hard trees & bst problem solved with the child + ancestor handling pattern. The best approach, one dfs (child + ancestor handling), runs in O(n) time and O(h) space. Below are 2 approaches in Java, from parent map + bfs up.

Problem

A fire starts at the node with value target. Each second it spreads to that node’s parent and children. Return the minimum time for the whole tree to burn.

Examples

Example 1

Input
root = [1, 2, 3, 4, 5, null, 6], target = 4
Output
4
Why
Second 1: 2. Second 2: 1 and 5. Second 3: 3. Second 4: 6.

Example 2

Input
root = [1, 2, 3], target = 1
Output
1

Constraints

  • The tree has 1 to 10^5 nodes; target exists.
  • Fire spreads from a burning node to its parent and children each second.

Animated walkthrough

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Concepts first, then the problem and every approach, step by step.

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Solutions

Try it yourself first. Then compare: each approach lists its idea, the steps, its time and space, and the Java code.

ApproachTimeSpace
Parent map + BFSO(n)O(n)
One DFS (child + ancestor handling)O(n)O(h)

1Parent map + BFS

TimeO(n)
SpaceO(n)

Record parents to treat the tree as an undirected graph. BFS from the target and count levels.

  1. Build a parent map and locate target.
  2. BFS over children and parent; count the levels after the first.
Java
class Solution {
    public static int minTime(TreeNode root, int target) {
        Map<TreeNode, TreeNode> parent = new HashMap<>();
        TreeNode src = null;
        Queue<TreeNode> q = new ArrayDeque<>();
        q.add(root);
        while (!q.isEmpty()) {
            TreeNode n = q.poll();
            if (n.val == target) src = n;
            if (n.left != null) { parent.put(n.left, n); q.add(n.left); }
            if (n.right != null) { parent.put(n.right, n); q.add(n.right); }
        }
        Set<TreeNode> burnt = new HashSet<>();
        q.add(src);
        burnt.add(src);
        int time = -1;
        while (!q.isEmpty()) {
            time++;
            for (int i = q.size(); i > 0; i--) {
                TreeNode n = q.poll();
                for (TreeNode nb : new TreeNode[] { n.left, n.right, parent.get(n) })
                    if (nb != null && burnt.add(nb)) q.add(nb);
            }
        }
        return time;
    }
}

2One DFS (child + ancestor handling)

TimeO(n)
SpaceO(h)

Return subtree height normally, but a negative distance when the subtree contains the target. At each ancestor, the fire reaches the far side after (distance to target) + (height of the other subtree) seconds. At the target itself, the answer is its own height below.

  1. At the target: best = max(best, height below); return -1.
  2. Ancestor with a negative side: best = max(best, dist + other height); return -(dist + 1).
  3. Otherwise return 1 + max(l, r).
Java
class Solution {
    private static int best;

    public static int minTime(TreeNode root, int target) {
        best = 0;
        dfs(root, target);
        return best;
    }

    private static int dfs(TreeNode n, int target) {
        if (n == null) return 0;
        int l = dfs(n.left, target), r = dfs(n.right, target);
        if (n.val == target) {
            best = Math.max(best, Math.max(l, r));
            return -1;
        }
        if (l >= 0 && r >= 0) return 1 + Math.max(l, r);
        int dist = -Math.min(l, r);
        int other = l >= 0 ? l : r;
        best = Math.max(best, dist + other);
        return -(dist + 1);
    }
}

Edge cases to test

  • Target is the root
  • Target is a leaf on the longest branch

Hints

Hint 1

Same problem as Amount of Time for Binary Tree to Be Infected: the answer is the farthest node's distance from the target.

FAQ

What is the best time complexity for Burning Tree?

One DFS (child + ancestor handling) runs in O(n) time and O(h) extra space.

Which pattern does Burning Tree use?

It is a trees & bst problem that uses the child + ancestor handling pattern. Other problems with the same pattern: Amount of Time for Binary Tree to Be Infected, All Nodes Distance K in Binary Tree.

Is there a brute force solution for Burning Tree?

Yes. Parent map + BFS takes O(n) time and O(n) space. Record parents to treat the tree as an undirected graph.

Which edge cases should I test for Burning Tree?

Target is the root; Target is a leaf on the longest branch.