I. INTRODUCTION The last two decades have been witness to intense activities in experimental and theo- retical study of correlation properties of interacting quantum systems. The problem is still challenging and open in the quantum many-body systems which are non-integrable. The task becomes simplified for integrable systems like one dimensional uniform Bose gas described by the Lieb-Liniger (LL) model [1,2], which assumes that particles interact via a δ -function repulsive potential. Since the experimental observation of Bose Einstein condensate (BEC) in ultracold trapped alkali atomic vapors, a lot of theoretical and experimental work has been done to study its correlation properties [3-12]. In the recent experimental situation, it is easy to achieve a quasi-one-dimensional strongly interacting degenerate Bose gas in a highly anisotropic trap [13-15] which is correctly described by the Lieb-Liniger model [16]. The complementary case is the weakly interacting (via finite range forces) trapped Bose gas. Although it is commonly believed that the Gross Pitaveskii (GP) equation is adequate for weakly interacting Bose gases, but a more rigorous and accurate many-body treatment, incorporating realistic interatomic interactions and interatomic correlations, is crucial for studying correlation properties in realistic condensates. The first motivation of the present work is to investigate the importance of finite size and trapping effects on the ground state correlation properties of ultracold atomic BECs. The second motivation of our study is to assess the validity of shape-independent approximation in the correlation properties. Dilute BECs are known to possess shape-independent (SI) property which is frequently described by the δ -function potential, whose strength is proportional to the s -wave scattering length ( a$_{s}$ ). In the mean-field description, the effective interaction is determined by the factor Na$_{s}$ (where a$_{s}$ is expressed in units of oscillator length of the trap), whereas in the full many-body description we solve the many-body Schr¨ odinger equation for different two-body potentials that generate identical a$_{s}$ . Thus, our present calculation serves as a stringent test of the SI approximation and verifies whether the long range correlation at all affects the correlation properties or not. Inclusion of all possible two-body correlations in our ab initio many-body method correctly calculates correlation effects at zero temperature. Use of the realistic van der Waals interatomic potential with a long attractive tail provides the realistic aspects of the correlation function and its momentum distribution. Specially the effect of short range repulsion in the interatomic potential is expected to be reflected in the 2