Breadth First Search (BFS) is started on a binary tree beginning from the root vertex. There is a vertex $t$ at a distance four from the root. If $t$ is the $n$-th vertex in this BFS traversal, then the maximum possible value of $n$ is _________.

Breadth First Search (BFS) on a tree traverses all vertices level by level, starting from the root at level $0$ (distance $0$).

The maximum number of nodes at any level $k$ in a binary tree is $2^k$.

To maximize the traversal index $n$ of a vertex $t$ located at distance $4$ (level $4$):

1. All previous levels ($0, 1, 2, 3$) must contain the maximum possible number of nodes.

2. The vertex $t$ must be visited as the very last node at level $4$.

The maximum number of nodes from level $0$ to level $4$ is:

Thus, the maximum possible value of $n$ is 31.