In the time-domain BEM with Haar wavelets for 2-D diffusion problems, the relation between the number of non-zero entries of the coefficient matrices and the degree of freedom (DOF) N is theoretically investigated using the information on the size and the arrangement of the support of the basis functions. The coefficient matrices are compressed using the Beylkin-type level-independent truncation scheme with a DOF-independent prescribed threshold value. The number of non-zero entries of the matrix G(L,p) and H(L,p) (1 ≤ p ≤ L, L: current time step), N(G(L,p)) and N(H(L,p)), increases in proportion to the factors log N, N1/2, N and N log N, except for the behavior in the smaller DOF range where N(G(L,p)) and N(H(L,p))∼O(N2). For M»1 and N»1, N(G(L,p)) and N(H(L,p)) show O(N log N) in matrix compression with a prescribed threshold value λ.