Researchers analyzing data collected from a stream network need to determine the data's second-order dependence structure before identifying an appropriate model and predicting the values of response variable at unsampled sites (via Kriging) or the average value of the variable over a stream interval (via Block Kriging). The Torgegram enables us to graphically detect the dependence structure of stream network data, but formal tests that resemble those in Euclidean space have not yet been developed on stream networks. The objective of this thesis is to construct nonparametric tests for pure tail-down and pure tail-up dependence on stream networks. These tests are based on the characteristics of some Torgegram components under specific types of dependence structure: the test for tail-down dependence relies on the fact that Type-0 and Type-1 subsemivariances at the same lag are equal when the covariance structure is tail-down, while the test for tail-up dependence takes advantage of a "flat" FUSD semivariogram under tail-up dependence.
Several test statistics are proposed to test for pure tail-down and tail-up dependence. The general form of these test statistics is a fraction, whose numerator is the sample size multiplied by the squared difference of specific semivariances (or subsemivariances) and denominator is a consistent estimator of the variance of the square root of the numerator. The asymptotic behaviors of the vectors of semivariances or subsemivariances are proved on regular rooted binary trees with an increasing number of levels, so that these test statistics converge in distribution to a chi-squared random variable with one degree of freedom. In order to obtain consistent estimators of the variance-covariance matrices of the vectors of semivariances or subsemivariances, two methods are introduced in this thesis: the plug-in method, which computes estimators as functions of linear combinations of FCSD and FUDJ semivariances, and the subsampling method, which computes estimators based on the semivariances or subsemivariances from overlapping subsamples. Then, those test statistics for tail-down and tail-up dependence are extended to irregular stream networks, based on Torgegram components and subsampling estimators. Simulation studies on regular rooted binary trees and real stream networks (from two real datasets) demonstrate the good performance of these tests: the larger the sample size or the stronger the spatial dependence, the higher the powers of the tests. However, the test for tail-down dependence requires Type-0 subsemivariances estimated from a sufficient number of pairs of sites on the same stream segments, which not every stream network dataset has.