<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
class Solution:
    def getDirections(self, root: Optional[TreeNode], startValue: int, destValue: int) -> str:

        #step1 find LCA
        lca = []


        def dfs(node):
            seenStart, seenDest = False, False
            if not node:
                return False, False
            if node.val == startValue:
                seenStart = True 
            elif node.val == destValue:
                seenDest = True 
            if seenStart and seenDest:
                lca.append(node)
                return False, False 
            leftStart, leftDest = dfs(node.left)
            rightStart, rightDest = dfs(node.right)
            seenStart = seenStart | leftStart | rightStart
            seenDest = seenDest | leftDest | rightDest
            if seenStart and seenDest:
                lca.append(node)
                return False, False
            return seenStart, seenDest

        dfs(root)

        res = [""]

        up = []

        def upPath(node):
            if not node:
                return False
            if node.val == startValue:
                res[0] = "".join(up)
                return True 
            up.append("U")
            l = upPath(node.left)
            if l:
                return True 
            up.pop()
            up.append("U")
            r = upPath(node.right)
            if r:
                return True
            up.pop()
            return False

        down = []

        def downPath(node):
            if not node:
                return False
            if node.val == destValue:
                res[0] += "".join(down)
                return True
            down.append("L")
            if downPath(node.left):
                return True

            down.pop()
            down.append("R")
            if downPath(node.right):
                return False
            down.pop()
            return 

        upPath(lca[0])
        downPath(lca[0])
        return res[0] 
        
            

        
            

        # parents = {}
        # startNode = []
        # def dfs(node, parent):
        #     if not node:
        #         return 
        #     if not startNode and node.val == startValue:
        #         startNode.append(node)
        #     parents[node.val] = parent 
        #     dfs(node.left, node)
        #     dfs(node.right, node)

        # dfs(root, TreeNode(val=-1))
        # seen = set()
        # results = []
        # path = []

        # def dfs2(node):
        #     if not node or node.val == -1:
        #         return False
        #     elif node.val in seen:
        #         return False
        #     if node.val == destValue:
        #         results.append("".join(path))
        #         return True
        #     seen.add(node.val)
        #     path.append("L")
        #     left = dfs2(node.left)
        #     if left:
        #         return True 
        #     path.pop()
        #     path.append("R")
        #     right = dfs2(node.right)
        #     if right:
        #         return True 
        #     path.pop()
        #     path.append("U")
        #     parent = dfs2(parents[node.val])
        #     if parent:
        #         return True
        #     path.pop()
        #     return False
        
        # dfs2(startNode[0])
        # return results[0]







        
        



56468