The paper introduces iSAM2, an incremental smoothing and mapping algorithm that uses a novel data structure called the Bayes tree. The Bayes tree provides a more efficient and accurate way to represent and update the square root information matrix used in simultaneous localization and mapping (SLAM). Unlike traditional clique trees, the Bayes tree is directed and maps more naturally to the square root information matrix of the SLAM problem. This allows for more efficient incremental updates and relinearization, eliminating the need for periodic batch steps. The Bayes tree is derived from the elimination of a factor graph, which is a graphical model used to represent probabilistic inference problems. The elimination process converts the factor graph into a Bayes net, which is then converted into a tree-structured graphical model. This tree-structured model allows for efficient optimization and marginalization. The Bayes tree is similar to a junction tree but is directed and better captures the equivalence between sparse linear algebra and inference in graphical models. The iSAM2 algorithm uses the Bayes tree to perform incremental reordering and just-in-time relinearization. This allows for efficient updates to the square root information matrix, which is used to compute the full map and trajectory at any time. The algorithm is tested on a range of real and simulated datasets, showing that it compares favorably with other recent mapping algorithms in both quality and efficiency. The paper also discusses the importance of variable ordering in the efficiency of the algorithm. An optimal variable ordering minimizes fill-in, which refers to additional entries in the square root information matrix that are created during the elimination process. The paper proposes an incremental variable ordering strategy that forces the most recently accessed variables to the end of the ordering, which can lead to more efficient updates. The iSAM2 algorithm is compared to other state-of-the-art SLAM algorithms, including iSAM1, HOG-Man, and SPA. The results show that iSAM2 achieves significant improvements in efficiency and accuracy, particularly in scenarios with large loop closures. The algorithm is also tested on a range of 2D and 3D datasets, including simulated and real-world data, showing its versatility and effectiveness in various SLAM applications.The paper introduces iSAM2, an incremental smoothing and mapping algorithm that uses a novel data structure called the Bayes tree. The Bayes tree provides a more efficient and accurate way to represent and update the square root information matrix used in simultaneous localization and mapping (SLAM). Unlike traditional clique trees, the Bayes tree is directed and maps more naturally to the square root information matrix of the SLAM problem. This allows for more efficient incremental updates and relinearization, eliminating the need for periodic batch steps. The Bayes tree is derived from the elimination of a factor graph, which is a graphical model used to represent probabilistic inference problems. The elimination process converts the factor graph into a Bayes net, which is then converted into a tree-structured graphical model. This tree-structured model allows for efficient optimization and marginalization. The Bayes tree is similar to a junction tree but is directed and better captures the equivalence between sparse linear algebra and inference in graphical models. The iSAM2 algorithm uses the Bayes tree to perform incremental reordering and just-in-time relinearization. This allows for efficient updates to the square root information matrix, which is used to compute the full map and trajectory at any time. The algorithm is tested on a range of real and simulated datasets, showing that it compares favorably with other recent mapping algorithms in both quality and efficiency. The paper also discusses the importance of variable ordering in the efficiency of the algorithm. An optimal variable ordering minimizes fill-in, which refers to additional entries in the square root information matrix that are created during the elimination process. The paper proposes an incremental variable ordering strategy that forces the most recently accessed variables to the end of the ordering, which can lead to more efficient updates. The iSAM2 algorithm is compared to other state-of-the-art SLAM algorithms, including iSAM1, HOG-Man, and SPA. The results show that iSAM2 achieves significant improvements in efficiency and accuracy, particularly in scenarios with large loop closures. The algorithm is also tested on a range of 2D and 3D datasets, including simulated and real-world data, showing its versatility and effectiveness in various SLAM applications.