Combinatorially composing Chebyshev polynomials

We present a combinatorial proof of two fundamental composition identities associated with Chebyshev polynomials. Namely, for all m , n ⩾ 0 , T m ( T n ( x ) ) = T mn ( x ) and U m − 1 ( T n ( x ) ) U n − 1 ( x ) = U mn − 1 ( x ) .