<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>

# Definition for a binary tree node.
# class TreeNode:
#     def __init__(self, val=0, left=None, right=None):
#         self.val = val
#         self.left = left
#         self.right = right
INF = 10**9

def getInvMap(root, out):
    if root == None:
        return
    out[root.val] = root
    getInvMap(root.left, out)
    getInvMap(root.right, out)

def buildPMap(root, out):
    if root == None:
        return
    if root.left != None:
        out[root.left] = root
        buildPMap(root.left, out)
    if root.right != None:
        out[root.right] = root
        buildPMap(root.right, out)

def buildDpthMap(root, dpth, out):
    if root == None:
        return
    out[root.val] = dpth
    buildDpthMap(root.left, dpth + 1, out)
    buildDpthMap(root.right, dpth + 1, out)
    
class Solution:
    def getDirections(self, root: Optional[TreeNode], startValue: int, destValue: int) -> str:
        invMap = {}
        getInvMap(root, invMap)
        pMap = {}
        buildPMap(root, pMap)
        dpthMap = {}
        buildDpthMap(root, 0, dpthMap)
        n = len(invMap)
        wlkr_start = invMap[startValue]
        d_start = dpthMap[startValue]
        wlkr_end = invMap[destValue]
        d_end = dpthMap[destValue]

        moves_start = []
        moves_end = []
        while d_start > d_end:
            wlkr_start = pMap[wlkr_start]
            moves_start.append('U')
            d_start -= 1
        while d_end > d_start:
            p = pMap[wlkr_end]
            if wlkr_end == p.left:
                moves_end.append('L')
            else:
                moves_end.append('R')
            wlkr_end = p
            d_end -= 1
        while wlkr_end != wlkr_start:
            wlkr_start = pMap[wlkr_start]
            moves_start.append('U')
            p = pMap[wlkr_end]
            moves_end.append('L' if wlkr_end == p.left else 'R')
            wlkr_end = p
        moves_end.reverse()
        return ''.join(moves_start) + ''.join(moves_end)

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