For any non-negative weighting on a (k,l)-bi-regular tree, the growth rate of all walks is determined by the growth rate of non-backtracking walks through Puder's conjectured formula.
and Peres, Y., Probability on trees and networks, Vol
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A note on Puder's generalised co-growth formula for trees
For any non-negative weighting on a (k,l)-bi-regular tree, the growth rate of all walks is determined by the growth rate of non-backtracking walks through Puder's conjectured formula.