  A foundational question in the theory of linear compartmental models is how
to assess whether a model is structurally identifiable -- that is, whether
parameter values can be inferred from noiseless data -- directly from the
combinatorics of the model. Our main result completely answers this question
for models (with one input and one output) in which the underlying graph is a
bidirectional tree; moreover, identifiability of such models can be verified
visually}. Models of this structure include two families of models often
appearing in biological applications: catenary and mammillary models. Our
analysis of such models is enabled by two supporting results, which are
significant in their own right. One result gives the first general formula for
the coefficients of input-output equations (certain equations that can be used
to determine identifiability) that allows for input and output to be in
distinct compartments}. In another supporting result, we prove that
identifiability is preserved when a model is enlarged and altered in specific
ways involving adding a new compartment with a bidirected edge to an existing
compartment.
