<p>You are given the <code>root</code> of a binary tree with <strong>unique</strong> values, and an integer <code>start</code>. At minute <code>0</code>, an <strong>infection</strong> starts from the node with value <code>start</code>.</p>

<p>Each minute, a node becomes infected if:</p>

<ul>
	<li>The node is currently uninfected.</li>
	<li>The node is adjacent to an infected node.</li>
</ul>

<p>Return <em>the number of minutes needed for the entire tree to be infected.</em></p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/06/25/image-20220625231744-1.png" style="width: 400px; height: 306px;" />
<pre>
<strong>Input:</strong> root = [1,5,3,null,4,10,6,9,2], start = 3
<strong>Output:</strong> 4
<strong>Explanation:</strong> The following nodes are infected during:
- Minute 0: Node 3
- Minute 1: Nodes 1, 10 and 6
- Minute 2: Node 5
- Minute 3: Node 4
- Minute 4: Nodes 9 and 2
It takes 4 minutes for the whole tree to be infected so we return 4.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/06/25/image-20220625231812-2.png" style="width: 75px; height: 66px;" />
<pre>
<strong>Input:</strong> root = [1], start = 1
<strong>Output:</strong> 0
<strong>Explanation:</strong> At minute 0, the only node in the tree is infected so we return 0.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li>The number of nodes in the tree is in the range <code>[1, 10<sup>5</sup>]</code>.</li>
	<li><code>1 &lt;= Node.val &lt;= 10<sup>5</sup></code></li>
	<li>Each node has a <strong>unique</strong> value.</li>
	<li>A node with a value of <code>start</code> exists in the tree.</li>
</ul>

class Solution {
public:
    TreeNode* lca(TreeNode* root, TreeNode* p, TreeNode* q) {
        if (!root || root == p || root == q) {
            return root;
        }
        TreeNode* left = lca(root->left, p, q);
        TreeNode* right = lca(root->right, p, q);
        if (!left) {
            return right;
        }
        if (!right) {
            return left;
        }
        return root;
    }
    TreeNode* k = nullptr;
    void getStart(TreeNode* root, int target) {
        queue<TreeNode*> q;
        q.push(root);
        while (!q.empty()) {
            int size = q.size();
            for (int i = 0; i < size; i++) {
                TreeNode* node = q.front();
                q.pop();
                if (node->val == target) {
                    k = node;
                    return;
                }
                if (node->left) {
                    q.push(node->left);
                }
                if (node->right) {
                    q.push(node->right);
                }
            }
        }
    }
    int findDistance(TreeNode* root, TreeNode* target) {
        queue<TreeNode*> q;
        q.push(root);
        int level = 0;
        while (!q.empty()) {
            int size = q.size();
            for (int i = 0; i < size; i++) {
                TreeNode* node = q.front();
                q.pop();
                if (node == target) {
                    return level;
                }
                if (node->left) {
                    q.push(node->left);
                }
                if (node->right) {
                    q.push(node->right);
                }
            }
            level++;
        }
        return -1;
    }
    int depth(TreeNode* root) {
        if (!root) {
            return 0;
        }
        int left = depth(root->left);
        int right = depth(root->right);
        return max(left, right) + 1;
    }
    int amountOfTime(TreeNode* root, int start) {
        queue<TreeNode*> q;
        q.push(root);
        getStart(root, start);
        int ans = 0;
        TreeNode* anc = lca(root, k, root);
        int f = findDistance(anc, k);
        int s = findDistance(anc, root);
        ans = max(ans, f + s);
        int dep_l = depth(root->left) + 1;
        int dep_r = depth(root->right) + 1;
        cout << dep_l << ' ' << dep_r;
        int level = 1;
        while (!q.empty()) {
            int size = q.size();
            for (int i = 0; i < size; i++) {
                TreeNode* node = q.front();
                q.pop();
                if (level == dep_l && (i == 0 || i == size - 1)) {
                    TreeNode* anc = lca(root, k, node);
                    f = findDistance(anc, k);
                    s = findDistance(anc, node);
                    ans = max(ans, f + s);
                }
                else if (level == dep_r && (i == 0 || i == size - 1)) {
                    TreeNode* anc = lca(root, k, node);
                    f = findDistance(anc, k);
                    s = findDistance(anc, node);
                    ans = max(ans, f + s);
                }
                if (node->left) {
                    q.push(node->left);
                }
                if (node->right) {
                    q.push(node->right);
                }
            }
            level++;
        }
        return ans;
    }
};
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