  We investigate finite-size scaling of genuine multisite entanglement in the
ground state of quantum spin-1/2 Heisenberg ladders. We obtain the ground
states of odd- and even-legged Heisenberg ladder Hamiltonians and compute
genuine multisite entanglement, the generalized geometric measure (GGM), which
shows that for even rungs, GGM increases for odd-legged ladder while it
decreases for even ones. Interestingly, the ground state obtained by
short-range dimer coverings, under the resonating valence bond (RVB) ansatz,
encapsulates the qualitative features of GGM for both the ladders. We find that
while the GGMs for higher legged odd- and even-ladders converge to a single
value in the asymptotic limit of a large number of rungs, the finite-size
scaling exponents of the same tend to diverge. The scaling exponent of GGM
obtained by employing density matrix recursion method is therefore a reliable
quantity in distinguishing the odd-even dichotomy in Heisenberg ladders, even
when the corresponding multisite entanglements merge.
