By-tree information gathering is an essential task in precision agriculture achieved by ground mobile sensors, but it can be time- and labor-intensive. In this paper we present an algorithmic framework to perform real-time and on-the-go detection of trees and key geometric characteristics (namely, width and height) with wheeled mobile robots in the field. Our method is based on the fusion of 2D domain-specific data (normalized difference vegetation index [NDVI] acquired via a red-green-near-infrared [RGN] camera) and 3D LiDAR point clouds, via a customized tree landmark association and parameter estimation algorithm. The proposed system features a multi-modal and entropy-based landmark correspondences approach, integrated into an underlying Kalman filter system to recognize the surrounding trees and jointly estimate their spatial and vegetation-based characteristics. Realistic simulated tests are used to evaluate our proposed algorithm's behavior in a variety of settings. Physical experiments in agricultural fields help validate our method's efficacy in acquiring accurate by-tree information on-the-go and in real-time by employing only onboard computational and sensing resources.