  For the exactly solvable model of exponential last passage percolation on
$\mathbb{Z}^2$, it is known that given any non-axial direction, all the
semi-infinite geodesics starting from points in $\mathbb{Z}^2$ in that
direction almost surely coalesce, thereby forming a geodesic tree which has
only one end. It is widely understood that the geodesic trees are important
objects in understanding the geometry of the LPP landscape. In this paper we
study several natural questions about these geodesic trees and their
intersections. In particular, we obtain optimal (up to constants) upper and
lower bounds for the (power law) tails of the height and the volume of the
backward sub-tree rooted at a fixed point. We also obtain bounds for the
probability that the sub-tree contains a specific vertex, e.g. the sub-tree in
the direction $(1,1)$ rooted at the origin contains the vertex $-(n,n)$, which
answers a question analogous to the well-known midpoint problem in the context
of semi-infinite geodesics. Furthermore, we obtain bounds for the probability
that a pair of intersecting geodesics both pass through a given vertex. These
results are interesting in their own right as well as useful in several other
applications.
