For distinct points x, y in an ultrametric space (X, d), we prove that the midset M(x, y) ={z: d(x, z) =d(y, z)} equals X(B(x, r)?B(y, r)) where r=d(x, y), hence is clopen. This yields a dendrogram formula |M(x, y)|=n-?(c i)-?(c j) via the representing tree. Our main results are: (1) an equilateral rigidity theorem—constant midset cardinality forces the space to be equilateral; (2) a subset S is midset-convex if and only if it meets at least three children of its least common ancestor; (3) the Haar measure formula µ(M(x, y)) = 1-2p-(k+1) in Z_p for d(x, y) = p^{-k}. We also characterize the Non-Empty Midset Property completely and recover the result that ultrametric spaces with the Unique Midpoint Property have at most three points.

Ultrametric space, midset, equidistant set, non-empty midset property, unique midpoint property, p-adic integers, representing tree, Haar measure, midset-convexity.