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.