  Binary Tree States (BTS) are states whose decomposition on a quantum register
basis formed by a set of qubits can be made sequentially. Such states sometimes
appear naturally in many-body systems treated in Fock space when a global
symmetry is imposed, like the total spin or particle number symmetries.
Examples are the Dicke states, the eigenstates of the total spin for a set of
particles having individual spin $1/2$, or states obtained by projecting a BCS
states onto particle number, also called projected BCS in small superfluid
systems. Starting from a BTS state described on the set of $n$ qubits or
orbitals, the entanglement entropy of any subset of $ k$ qubits is analyzed.
Specifically, a practical method is developed to access the $k$
qubits/particles von Neumann entanglement entropy of the subsystem of interest.
Properties of these entropies are discussed, including scaling properties,
upper bounds, or how these entropies correlate with fluctuations. Illustrations
are given for the Dicke state and the projected BCS states.
