Let $\text{path(node1, node2)}$ denote the path from $\text{node1}$ to $\text{node2}$.

First consider the case where $\text{path(start, end)}$ goes through the root. Let's split this into $\text{path(start, root)} + \text{path(root, end)}$. We can perform a DFS (depth first search) to get $\text{path(root, end)}$. This path consists of 'L's and 'R's. We can do another DFS to get $\text{path(root, start)}$. Replacing the 'L's and 'R's of $\text{path(root, start)}$ with 'U's gives us $\text{path(start, root)}$. Now we can concatenate $\text{path(start, root)}$ and $\text{path(root, end)}$ to get the answer.

In the general case, $\text{path(start, end)}$ may not go through the root. Notice that this path goes up a non-negative number of times ($\verb|"U"|$s) before going down a non-negative number of times ($\verb|"L"|$s or $\verb|"R"|$s). The highest node in this path is known as the LCA (lowest common ancestor) of $\text{start}$ and $\text{end}$.

Here, $\text{path(root, start)} = \verb|"RRLLR"|$ and $\text{path(root, end)} = \verb|"RRRL"|$. Let's remove their longest common prefix, which is $\verb|"RR"|$. We have $\text{path(LCA, start)} = \verb|"LLR"|$ and $\text{path(LCA, end)} = \verb|"RL"|$.

Then we replace all characters in $\text{path(root, start)}$ with $\verb|"U"|$s to obtain $\text{path(start, lca)} = \verb|"UUU"|$. Finally, we get $\text{path(start, end)} = \text{path(start, LCA)} + \text{path(LCA, end)} = \verb|"UUU"| + \verb|"RL"| = \verb|"UUURL"|$.

Time complexity

Each DFS takes $\mathcal{O}(n)$ and our string operations never happen on strings exceeding length $\mathcal{O}(n)$. The time complexity is $\mathcal{O}(n)$.

Space complexity

The strings never exceed length $\mathcal{O}(n)$. The space complexity is $\mathcal{O}(n)$.

C++ Solution

/**
 * Definition for a binary tree node.
 * struct TreeNode {
 *     int val;
 *     TreeNode *left;
 *     TreeNode *right;
 *     TreeNode() : val(0), left(nullptr), right(nullptr) {}
 *     TreeNode(int x) : val(x), left(nullptr), right(nullptr) {}
 *     TreeNode(int x, TreeNode *left, TreeNode *right) : val(x), left(left),
 * right(right) {}
 * };
 */
class Solution {
public:
    void getPath(TreeNode *cur, int targetValue, string &path, string &ans) {
        if (!cur)
            return;
        if (cur->val == targetValue)
            ans = path;
        path.push_back('L');
        getPath(cur->left, targetValue, path, ans);
        path.back() = 'R';
        getPath(cur->right, targetValue, path, ans);
        path.pop_back();
    }

    string getDirections(TreeNode *root, int startValue, int destValue) {
        string tmpPath, startPath, destPath;
        getPath(root, startValue, tmpPath, startPath);
        getPath(root, destValue, tmpPath, destPath);
        // Find the first point at which the paths diverge
        auto [itr1, itr2] = mismatch(startPath.begin(), startPath.end(),
                                     destPath.begin(), destPath.end());
        return string(startPath.end() - itr1, 'U') +
               string(itr2, destPath.end());
    }
};

Java Solution

/**
 * Definition for a binary tree node.
 * public class TreeNode {
 *     int val;
 *     TreeNode left;
 *     TreeNode right;
 *     TreeNode() {}
 *     TreeNode(int val) { this.val = val; }
 *     TreeNode(int val, TreeNode left, TreeNode right) {
 *         this.val = val;
 *         this.left = left;
 *         this.right = right;
 *     }
 * }
 */
class Solution {
    // We use StringBuilder instead of String because String is immutable,
    // so appending a character takes O(length of string).
    // ans is a String[1] instead of String because in Java, arrays are passed by reference.
    void getPath(TreeNode cur, int targetValue, StringBuilder path, String[] ans) {
        if (cur == null)
            return;
        if (cur.val == targetValue)
            ans[0] = path.toString();
        int strLen = path.length();
        path.append("L");
        getPath(cur.left, targetValue, path, ans);
        path.replace(strLen, strLen+1, "R");
        getPath(cur.right, targetValue, path, ans);
        path.delete(strLen, strLen+1);
    }
    public String getDirections(TreeNode root, int startValue, int destValue) {
        StringBuilder tmpPath = new StringBuilder();
        String[] startPath = {""}, destPath = {" "};
        // Find the first point at which the paths diverge
        getPath(root, startValue, tmpPath, startPath);
        getPath(root, destValue, tmpPath, destPath);
        int i = 0;
        while (i < Math.min(startPath[0].length(), destPath[0].length()) && startPath[0].charAt(i) == destPath[0].charAt(i))
            i++;
        return "U".repeat(startPath[0].length()-i) + destPath[0].substring(i);
    }
}

Python Solution

# 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:
        # ans is a list so that it passes by reference
        def getPath(cur, targetValue, path, ans):
            if cur is None:
                return
            if cur.val == targetValue:
                ans.append(''.join(path));
            path.append('L');
            getPath(cur.left, targetValue, path, ans)
            path[-1] = 'R'
            getPath(cur.right, targetValue, path, ans)
            path.pop(-1)

        tmpPath = []
        startPath = []
        destPath = []
        getPath(root, startValue, tmpPath, startPath)
        getPath(root, destValue, tmpPath, destPath)
        startPath = startPath[0]
        destPath = destPath[0]

        # Find the first point at which the paths diverge
        i = 0
        while i < min(len(startPath), len(destPath)) and startPath[i] == destPath[i]:
            i += 1
        return 'U'*(len(startPath)-i) + destPath[i:]