<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:
        graph = defaultdict(list)
        
        def build_graph(node, parent=None):
            if node:
                if parent:
                    graph[node.val].append((parent.val, 'U'))
                    graph[parent.val].append((node.val, 'L' if node == parent.left else 'R'))
                if node.left:
                    build_graph(node.left, node)
                if node.right:
                    build_graph(node.right, node)
        
        build_graph(root)
        
        def find_path(src, dst):
            queue = [(src, "")]
            visited = set()
            
            while queue:
                node, path = queue.pop(0)
                
                if node == dst:
                    return path
                
                visited.add(node)
                
                for neighbor, direction in graph[node]:
                    if neighbor not in visited:
                        queue.append((neighbor, path + direction))
            
            return ""
        
        return find_path(startValue, destValue)

        
        def dfs(node):
            if not node:
                return

            if node.left:
                build_graph(node.left, node)
            if node.right:
                build_graph(node.right, node)
            
            dfs(node.left)
            dfs(node.right)

        dfs(root)
    
        def find_path(src, path, dis):
            visited = set([src])
           
            l, r, u = graph[src]
            if l == destValue:
                return shortest_path + "L", dis
            if r == destValue:
                return path + "R", dis
            if u == destValue:
                return path + "U", dis
            
            dis += 1
            if l and l not in visited:
                find_path(l, path + "L", dis)
            if r and r not in visited:
                find_path(r, path + "R", dis)
            if u and u not in visited:
                find_path(u, path + "U", dis)

        print(graph)
        
        l, r, u = graph[startValue]
        left_dis, right_dis, up_dis = float('inf'), float('inf'), float('inf')
        if l:
            left, left_dis = find_path(l, "L", 1)
        if r:
            right, right_dis = find_path(r, "R", 1)
        if u:
            up, up_dis = find_path(u, "U", 1)

        min_dis = min(left_dis, right_dis, up_dis)
        if min_dis == left_dis:
            return left
        elif min_dis == right_dis:
            return right
        else:
            return up
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