"content":"<p>There is an undirected tree with <code>n</code> nodes labeled from <code>0</code> to <code>n - 1</code>, rooted at node <code>0</code>. You are given a 2D integer array <code>edges</code> of length <code>n - 1</code> where <code>edges[i] = [a<sub>i</sub>, b<sub>i</sub>]</code> indicates that there is an edge between nodes <code>a<sub>i</sub></code> and <code>b<sub>i</sub></code> in the tree.</p>\n\n<p>At every node <code>i</code>, there is a gate. You are also given an array of even integers <code>amount</code>, where <code>amount[i]</code> represents:</p>\n\n<ul>\n\t<li>the price needed to open the gate at node <code>i</code>, if <code>amount[i]</code> is negative, or,</li>\n\t<li>the cash reward obtained on opening the gate at node <code>i</code>, otherwise.</li>\n</ul>\n\n<p>The game goes on as follows:</p>\n\n<ul>\n\t<li>Initially, Alice is at node <code>0</code> and Bob is at node <code>bob</code>.</li>\n\t<li>At every second, Alice and Bob <b>each</b> move to an adjacent node. Alice moves towards some <strong>leaf node</strong>, while Bob moves towards node <code>0</code>.</li>\n\t<li>For <strong>every</strong> node along their path, Alice and Bob either spend money to open the gate at that node, or accept the reward. Note that:\n\t<ul>\n\t\t<li>If the gate is <strong>already open</strong>, no price will be required, nor will there be any cash reward.</li>\n\t\t<li>If Alice and Bob reach the node <strong>simultaneously</strong>, they share the price/reward for opening the gate there. In other words, if the price to open the gate is <code>c</code>, then both Alice and Bob pay <code>c / 2</code> each. Similarly, if the reward at the gate is <code>c</code>, both of them receive <code>c / 2</code> each.</li>\n\t</ul>\n\t</li>\n\t<li>If Alice reaches a leaf node, she stops moving. Similarly, if Bob reaches node <code>0</code>, he stops moving. Note that these events are <strong>independent</strong> of each other.</li>\n</ul>\n\n<p>Return<em> the <strong>maximum</strong> net income Alice can have if she travels towards the optimal leaf node.</em></p>\n\n<p> </p>\n<p><strong class=\"example\">Example 1:</strong></p>\n<img alt=\"\" src=\"https://assets.leetcode.com/uploads/2022/10/29/eg1.png\" style=\"width: 275px; height: 275px;\" />\n<pre>\n<strong>Input:</strong> edges = [[0,1],[1,2],[1,3],[3,4]], bob = 3, amount = [-2,4,2,-4,6]\n<strong>Output:</strong> 6\n<strong>Explanation:</strong> \nThe above diagram represents the given tree. The game goes as follows:\n- Alice is initially on node 0, Bob on node 3. They open the gates of their respective nodes.\n Alice's net income is now -2.\n- Both Alice and Bob move to node 1. \n Since they reach here simultaneously, they open the gate together and share the reward.\n Alice's net income becomes -2 + (4 / 2) = 0.\n- Alice moves on to node 3. Since Bob already opened its gate, Alice's income remains unchanged.\n Bob moves on to node 0, and stops moving.\n- Alice moves on to node 4 and opens the gate there. Her net income becomes 0 + 6 = 6.\nNow, neither Alice nor Bob can make any further moves, and the game ends.\nIt is not possible for Alice to get a higher net income.\n</pre>\n\n<p><strong class=\"example\">Example 2:</strong></p>\n<img alt=\"\" src=\"https://assets.leetcode.com/uploads/2022/10/29/eg2.png\" style=\"width: 250px; height: 78px;\"/>\n<pre>\n<strong>Input:</strong>edges=[[0,1]],bob=1,amount=[-7280,2350]\n<strong>Output:</strong>-7280\n<strong>Explanation:</strong>\nAlicefollowsthepath0->1whereasBobfollowsthepath1->0.\nThus,Aliceopensthegateatnode0only.Hence,hernetincomeis-7280.\n</pre>\n\n<p> </p>\n<p><strong>Constraints:</strong></p>\n\n<ul>\n\t<li><code>2<=n<=10<sup>5</sup></code></li>\n\t<li><code>edges.length==n-1</code></li>\n\t<li><code>edges[i].length==2</code></li>\n\t<li><code>0<=a<sub>i</sub>,b<sub>i</sub><n</code></li>\n\t<li><code>a<sub>i</sub>!=b<sub>i</sub></code></li>\n\t<li><code>edges</code>representsa
"envInfo":"{\"cpp\": [\"C++\", \"<p>Compiled with <code> clang 11 </code> using the latest C++ 20 standard.</p>\\r\\n\\r\\n<p>Your code is compiled with level two optimization (<code>-O2</code>). <a href=\\\"https://github.com/google/sanitizers/wiki/AddressSanitizer\\\" target=\\\"_blank\\\">AddressSanitizer</a> is also enabled to help detect out-of-bounds and use-after-free bugs.</p>\\r\\n\\r\\n<p>Most standard library headers are already included automatically for your convenience.</p>\"], \"java\": [\"Java\", \"<p><code>OpenJDK 17</code>. Java 8 features such as lambda expressions and stream API can be used. </p>\\r\\n\\r\\n<p>Most standard library headers are already included automatically for your convenience.</p>\\r\\n<p>Includes <code>Pair</code> class from https://docs.oracle.com/javase/8/javafx/api/javafx/util/Pair.html.</p>\"], \"python\": [\"Python\", \"<p><code>Python 2.7.12</code>.</p>\\r\\n\\r\\n<p>Most libraries are already imported automatically for your convenience, such as <a href=\\\"https://docs.python.org/2/library/array.html\\\" target=\\\"_blank\\\">array</a>, <a href=\\\"https://docs.python.org/2/library/bisect.html\\\" target=\\\"_blank\\\">bisect</a>, <a href=\\\"https://docs.python.org/2/library/collections.html\\\" target=\\\"_blank\\\">collections</a>. If you need more libraries, you can import it yourself.</p>\\r\\n\\r\\n<p>For Map/TreeMap data structure, you may use <a href=\\\"http://www.grantjenks.com/docs/sortedcontainers/\\\" target=\\\"_blank\\\">sortedcontainers</a> library.</p>\\r\\n\\r\\n<p>Note that Python 2.7 <a href=\\\"https://www.python.org/dev/peps/pep-0373/\\\" target=\\\"_blank\\\">will not be maintained past 2020</a>. For the latest Python, please choose Python3 instead.</p>\"], \"c\": [\"C\", \"<p>Compiled with <code>gcc 8.2</code> using the gnu11 standard.</p>\\r\\n\\r\\n<p>Your code is compiled with level one optimization (<code>-O1</code>). <a href=\\\"https://github.com/google/sanitizers/wiki/AddressSanitizer\\\" target=\\\"_blank\\\">AddressSanitizer</a> is also enabled to help detect out-of-bounds and use-after-free bugs.</p>\\r\\n\\r\\n<p>Most standard library headers are already included automatically for your convenience.</p>\\r\\n\\r\\n<p>For hash table operations, you may use <a href=\\\"https://troydhanson.github.io/uthash/\\\" target=\\\"_blank\\\">uthash</a>. \\\"uthash.h\\\" is included by default. Below are some examples:</p>\\r\\n\\r\\n<p><b>1. Adding an item to a hash.</b>\\r\\n<pre>\\r\\nstruct hash_entry {\\r\\n int id; /* we'll use this field as the key */\\r\\n char name[10];\\r\\n UT_hash_handle hh; /* makes this structure hashable */\\r\\n};\\r\\n\\r\\nstruct hash_entry *users = NULL;\\r\\n\\r\\nvoid add_user(struct hash_entry *s) {\\r\\n HASH_ADD_INT(users, id, s);\\r\\n}\\r\\n</pre>\\r\\n</p>\\r\\n\\r\\n<p><b>2. Looking up an item in a hash:</b>\\r\\n<pre>\\r\\nstruct hash_entry *find_user(int user_id) {\\r\\n struct hash_entry *s;\\r\\n HASH_FIND_INT(users, &user_id, s);\\r\\n return s;\\r\\n}\\r\\n</pre>\\r\\n</p>\\r\\n\\r\\n<p><b>3. Deleting an item in a hash:</b>\\r\\n<pre>\\r\\nvoid delete_user(struct hash_entry *user) {\\r\\n HASH_DEL(users, user); \\r\\n}\\r\\n</pre>\\r\\n</p>\"], \"csharp\": [\"C#\", \"<p><a href=\\\"https://learn.microsoft.com/en-us/dotnet/csharp/whats-new/csharp-10\\\" target=\\\"_blank\\\">C# 10 with .NET 6 runtime</a></p>\"], \"javascript\": [\"JavaScript\", \"<p><code>Node.js 16.13.2</code>.</p>\\r\\n\\r\\n<p>Your code is run with <code>--harmony</code> flag, enabling <a href=\\\"http://node.green/\\\" target=\\\"_blank\\\">new ES6 features</a>.</p>\\r\\n\\r\\n<p><a href=\\\"https://lodash.com\\\" target=\\\"_blank\\\">lodash.js</a> library is included by default.</p>\\r\\n\\r\\n<p>For Priority Queue / Queue data structures, you may use 5.3.0 version of <a href=\\\"https://github.com/datastructures-js/priority-queue/tree/fb4fdb984834421279aeb081df7af624d17c2a03\\\" target=\\\"_blank\\\">datastructures-js/priority-queue</a> and 4.2.1 version of <a href=\\\"https://githu
