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The purpose of this paper is to extend the definition of Frechet distancewhich measures the distance between two curves to a distance (Frechet-Likedistance) which measures the similarity between two rooted trees. Thedefinition of Frechet-Like distance is as follows: Tow men start from the rootsof two trees. When they reach to a node with the degree of more than $2$, theyconstruct $k-1$ men which $k$ is the outgoing degree of the node and each manmonitor a man in another tree (there is a rope between them). The distance isthe minimum length of the ropes between the men and the men whom are monitoredand they all go forward (the geodesic distance between them to the root of thetree increases) and reach to the leaves of the trees. Here, I prove that theFrechet-Like distance between two trees is SNP-hard to compute.  I modify the definition of Frechet-Like distance to measure the distancebetween tow merge trees, and I prove the relation between the interleavingdistance and the modified Frechet-Like distance.

Frechet-Like Distances between Two Merge Trees