Solution
Approach 1: Paint Deepest Nodes
Intuition
We try a straightforward approach that has two phases.
The first phase is to identify the nodes of the tree that are deepest. To do this, we have to annotate the depth of each node. We can do this with a depth first search.
Afterwards, we will use that annotation to help us find the answer:

If the
node
in question has maximum depth, it is the answer. 
If both the left and right child of a
node
have a deepest descendant, then the answer is this parentnode
. 
Otherwise, if some child has a deepest descendant, then the answer is that child.

Otherwise, the answer for this subtree doesn't exist.
Algorithm
In the first phase, we use a depth first search dfs
to annotate our nodes.
In the second phase, we also use a depth first search answer(node)
, returning the answer for the subtree at that node
, and using the rules above to build our answer from the answers of the children of node
.
Note that in this approach, the answer
function returns answers that have the deepest nodes of the entire tree, not just the subtree being considered.
Complexity Analysis

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

Space Complexity: .
Approach 2: Recursion
Intuition
We can combine both depth first searches in Approach #1 into an approach that does both steps in one pass. We will have some function dfs(node)
that returns both the answer for this subtree, and the distance from node
to the deepest nodes in this subtree.
Algorithm
The Result
(on some subtree) returned by our (depthfirst search) recursion will have two parts:
Result.node
: the largest depth node that is equal to or an ancestor of all the deepest nodes of this subtree.
Result.dist
: the number of nodes in the path from the root of this subtree, to the deepest node in this subtree.
We can calculate these answers disjointly for dfs(node)
:

To calculate the
Result.node
of our answer:
If one
childResult
has deeper nodes, thenchildResult.node
will be the answer. 
If they both have the same depth nodes, then
node
will be the answer.


The
Result.dist
of our answer is always 1 more than the largestchildResult.dist
we have.
Complexity Analysis

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

Space Complexity: .
Analysis written by: @awice.