We propose nonparametric regression and density estimators on a network. A network is defined as a collection of edges that are connected by vertices. There are numerous types of networks, such as streets, subway lines, electrical wires, airline routes, or nerve fibers and blood vessels. While broadly applicable, our methodology focuses on the challenging cases in which the best estimator near a vertex depends on the amount of smoothness at the vertex. To estimate the function in a neighborhood of a vertex, a two-step procedure is proposed. The first step of this pretest estimator fits a separate local polynomial regression on each edge, and then tests for equality of the estimates at the vertex. If the null hypothesis is not rejected, the second step re-estimates the function in a small neighborhood of the vertex, subject to a joint equality constraint. Since the derivative of the function may be discontinuous at the vertex, a piecewise polynomial local regression estimate is used to model the change in slope. Our approach removes the bias near a vertex that has been noted for existing methods, which typically do not allow for discontinuity at vertices. The implementation of our approach is fast, the computation time scales only with data sub-linearly. We use the model to estimate the desntiy of spines on a dendritic tree. Despite the simple intuition and easy implementation of the two-step procedure, it leaves the significance level of the test a tuning parameter and the type II error of test is difficult to analyse. As an alternative approach, we minimise the MSE of the estimator near the vertex by penalising the $l_2$ norm of the jumps at the vertex. We propose a method for estimating the locally optimal bandwidth and penalty parameter, and derive their error bounds. In order to derive the rate of convergence of the selector, we develop a uniform almost sure asymptotic theory of our model. The theory is also of interest on its own right. We show that the uniform almost sure asymptotic theory holds with probability $1$, uniformly over the neighborhood of a vertex, bandwidth and penalty parameters. We apply our model to New York City taxi data and study how the average trip cost for a taxi ride changes with trip origin along the streets of Manhattan. Finally we study the asymptotic properties of the penalized estimator with $l_r$ penalty, for $r>0$. We derive the limiting distributions of the estimates, and show that, under appropriate conditions, the limiting distributions of the jumps put positive probability mass at zero, so we can obtain continuous estimates at the vertex. We develop a bootstrap estimator of the bias and variance of the proposed penalised estimator, and justify it using asymptotic arguments.