We study a class of fused lasso problems where the estimated parameters in a sequence are regressed toward their respective observed values (fidelity loss), with $\ell_1$ norm penalty (regularization loss) on the differences between successive parameters, which promotes local constancy. In many applications, there is a coefficient, often denoted as $λ$, on the regularization term, which adjusts the relative importance between the two losses. In this paper, we characterize how the optimal solution evolves with the increment of $λ$. We show that, if all fidelity loss functions are convex piecewise linear, the optimal value for \emph{each} variable changes at most $O(nq)$ times for a problem of $n$ variables and total $q$ breakpoints. On the other hand, we present an algorithm that solves the path of solutions of \emph{all} variables in $\tilde{O}(nq)$ time for all $λ\geq 0$. Interestingly, we find that the path of solutions for each variable can be divided into up to $n$ locally convex-like segments. For problems of arbitrary convex loss functions, for a given solution accuracy, one can transform the loss functions into convex piecewise linear functions and apply the above results, giving pseudo-polynomial bounds as $q$ becomes a pseudo-polynomial quantity. To our knowledge, this is the first work to solve the path of solutions for fused lasso of non-quadratic fidelity loss functions.