<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>

class Solution:
    def __init__(self):
        self.parent = {}
        self.height = {}
    
    def getDirections(self, root: Optional[TreeNode], startValue: int, destValue: int) -> str:
        firstPath = []
        secondPath = []
        
        lca = self.findLCA(root, startValue, destValue)
        self.construct(lca, firstPath, startValue)
        self.construct(lca, secondPath, destValue)
        
        firstPath.reverse()
        secondPath.reverse()

        return 'U' * len(firstPath) + ''.join(secondPath)
        
    def dfs(self, p, root, h):
        if not root:
            return
        self.dfs(root, root.left, h + 1)
        self.dfs(root, root.right, h + 1)
        self.parent[root.val] = p
        self.height[root.val] = h
    
    def findLCA(self, root, u, v):
        self.dfs(None, root, 0)
        while u != v:
            if self.parent[u] == self.parent[v]:
                return self.parent[u]
            if self.height[u] > self.height[v]:
                if self.parent[u].val == v:
                    return self.parent[u]
                u = self.parent[u].val
            else:
                if self.parent[v].val == u:
                    return self.parent[v]
                v = self.parent[v].val
        
        return None
    
    def construct(self, root, s, val):
        if not root:
            return False
        if root.val == val:
            return True
        
        left = self.construct(root.left, s, val)
        right = self.construct(root.right, s, val)
        
        if left:
            s.append('L')
        elif right:
            s.append('R')
        
        return left or right
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