  By considering the continuum scaling limit of the $A_{4}$ RSOS lattice model
of Andrews-Baxter-Forrester with integrable boundaries, we derive excited state
TBA equations describing the boundary flows of the tricritical Ising model.
Fixing the bulk weights to their critical values, the integrable boundary
weights admit a parameter $\xi $ which plays the role of the perturbing
boundary field $\phi_{1,3}$ and induces the renormalization group flow between
boundary fixed points. The boundary TBA equations determining the RG flows are
derived in the $\mathcal{B}_{(1,2)}\to \mathcal{B}_{(2,1)}$ example. The
induced map between distinct Virasoro characters of the theory are specified in
terms of distribution of zeros of the double row transfer matrix.
