For a given binary tree with a specified root, where the depth of each node is defined as the shortest distance to the root, find and return the smallest subtree that encompasses all the deepest nodes in the original tree. A node is considered the deepest if it holds the maximum depth among all nodes in the entire tree. The subtree of a node is a tree composed of that node and all its descendants. Resolve this problem using the Euler tour method.

For a given binary tree with a specified root, where the depth of each node is defined as the shortest distance to the root, find and return the smallest subtree that encompasses all the deepest nodes in the original tree. A node is considered the deepest if it holds the maximum depth among all nodes in the entire tree. The subtree of a node is a tree composed of that node and all its descendants. Resolve this problem using the Euler tour method