To my knowledge from papers I have read, the number of pods is the main parameter used to determine the number of switches at each level along with the number of nodes. For instance, in each pod/k in fat-tree topology, there are k core switches, K/2 aggregation switches, K/2 edge switches and (k/2)^2 nodes/hosts. So my question is can we call a network topology that has more number of nodes (e.g. k*2) as a fat tree topology?


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You appear to have a rather rigid definition for a fat tree. Generally however, a fat tree is any tree where the links get fatter the more you approach the core (or the root).

A skinny tree is a tree where all links have the same bandwidth/speed. Any tree that has some speed increase towards the root can be called fat.

Fat trees that have no oversubscribed uplinks and are therefore impossible to be congested are rather rare - you'd need a 50 Gbit/s uplink for a branch of 50 ports with 1 Gbit/s each.

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