## Solution

#### Approach 1: Annotate Parent

Intuition

If we know the parent of every node x, we know all nodes that are distance 1 from x. We can then perform a breadth first search from the target node to find the answer.

Algorithm

We first do a depth first search where we annotate every node with information about it's parent.

After, we do a breadth first search to find all nodes a distance K from the target.

Complexity Analysis

• Time Complexity: , where is the number of nodes in the given tree.

• Space Complexity: .

#### Approach 2: Percolate Distance

Intuition

From root, say the target node is at depth 3 in the left branch. It means that any nodes that are distance K - 3 in the right branch should be added to the answer.

Algorithm

Traverse every node with a depth first search dfs. We'll add all nodes x to the answer such that node is the node on the path from x to target that is closest to the root.

To help us, dfs(node) will return the distance from node to the target. Then, there are 4 cases:

• If node == target, then we should add nodes that are distance K in the subtree rooted at target.

• If target is in the left branch of node, say at distance L+1, then we should look for nodes that are distance K - L - 1 in the right branch.

• If target is in the right branch of node, the algorithm proceeds similarly.

• If target isn't in either branch of node, then we stop.

In the above algorithm, we make use of the auxillary function subtree_add(node, dist) which adds the nodes in the subtree rooted at node that are distance K - dist from the given node.

Complexity Analysis

• Time Complexity: , where is the number of nodes in the given tree.

• Space Complexity: .

Analysis written by: @awice.