<p>You are given the <code>root</code> of a <strong>binary tree</strong> with <code>n</code> nodes. Each node is uniquely assigned a value from <code>1</code> to <code>n</code>. You are also given an integer <code>startValue</code> representing the value of the start node <code>s</code>, and a different integer <code>destValue</code> representing the value of the destination node <code>t</code>.</p>

<p>Find the <strong>shortest path</strong> starting from node <code>s</code> and ending at node <code>t</code>. Generate step-by-step directions of such path as a string consisting of only the <strong>uppercase</strong> letters <code>&#39;L&#39;</code>, <code>&#39;R&#39;</code>, and <code>&#39;U&#39;</code>. Each letter indicates a specific direction:</p>

<ul>
	<li><code>&#39;L&#39;</code> means to go from a node to its <strong>left child</strong> node.</li>
	<li><code>&#39;R&#39;</code> means to go from a node to its <strong>right child</strong> node.</li>
	<li><code>&#39;U&#39;</code> means to go from a node to its <strong>parent</strong> node.</li>
</ul>

<p>Return <em>the step-by-step directions of the <strong>shortest path</strong> from node </em><code>s</code><em> to node</em> <code>t</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/11/15/eg1.png" style="width: 214px; height: 163px;" />
<pre>
<strong>Input:</strong> root = [5,1,2,3,null,6,4], startValue = 3, destValue = 6
<strong>Output:</strong> &quot;UURL&quot;
<strong>Explanation:</strong> The shortest path is: 3 &rarr; 1 &rarr; 5 &rarr; 2 &rarr; 6.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/11/15/eg2.png" style="width: 74px; height: 102px;" />
<pre>
<strong>Input:</strong> root = [2,1], startValue = 2, destValue = 1
<strong>Output:</strong> &quot;L&quot;
<strong>Explanation:</strong> The shortest path is: 2 &rarr; 1.
</pre>

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

<ul>
	<li>The number of nodes in the tree is <code>n</code>.</li>
	<li><code>2 &lt;= n &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= Node.val &lt;= n</code></li>
	<li>All the values in the tree are <strong>unique</strong>.</li>
	<li><code>1 &lt;= startValue, destValue &lt;= n</code></li>
	<li><code>startValue != destValue</code></li>
</ul>

from collections import deque
from dataclasses import dataclass


@dataclass
class Step:
    value: int
    direction: str

    def __eq__(self, other) -> bool:
        return (self.value == other.value) and (self.direction == other.direction)


class RootCenteredSolver:
    def __init__(self, root: TreeNode):
        self.root = root
        self.path = deque([])

    def reset_path(self):
        self.path = deque([])

    def search(self, node: TreeNode, target: int) -> bool:
        if node.val == target:
            return True
        if node.left:
            self.path.append(Step(node.val, "L"))
            if self.search(node.left, target):
                return True
            self.path.pop()
        if node.right:
            self.path.append(Step(node.val, "R"))
            if self.search(node.right, target):
                return True
            self.path.pop()
        return False

    def find_path(self, target: int):
        self.search(self.root, target)
        path_to_target = self.path
        self.reset_path()
        return list(path_to_target)

    def infer_directions(self, src_path, dst_path) -> str:
        i = 0
        for src_step, dst_step in zip(src_path, dst_path):
            if src_step != dst_step:
                break
            i += 1
        dirs = "U" * (len(src_path) - i)
        dirs += "".join([step.direction for step in dst_path[i:]])
        return dirs

    def solve(self, src: int, dst: int) -> str:
        src_path = self.find_path(src)
        dst_path = self.find_path(dst)
        dirs = self.infer_directions(src_path, dst_path)
        return dirs


class Solution:
    def getDirections(
        self, 
        root: Optional[TreeNode], 
        startValue: int, 
        destValue: int,
    ) -> str:
        solver = RootCenteredSolver(root)
        return solver.solve(startValue, destValue)

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