diff --git a/Sampling_based_Planning/rrt_2D/batch_informed_trees.py b/Sampling_based_Planning/rrt_2D/batch_informed_trees.py
index e69de29..8627cc9 100644
--- a/Sampling_based_Planning/rrt_2D/batch_informed_trees.py
+++ b/Sampling_based_Planning/rrt_2D/batch_informed_trees.py
@@ -0,0 +1,278 @@
+"""
+Batch Informed Trees (BIT*)
+@author: huiming zhou
+"""
+
+import os
+import sys
+import math
+import random
+import copy
+import numpy as np
+import matplotlib.pyplot as plt
+import matplotlib.patches as patches
+
+sys.path.append(os.path.dirname(os.path.abspath(__file__)) +
+                "/../../Sampling_based_Planning/")
+
+from Sampling_based_Planning.rrt_2D import env, plotting, utils
+
+
+class Node:
+    def __init__(self, x, y):
+        self.x = x
+        self.y = y
+        self.parent = None
+
+
+class Tree:
+    def __init__(self, x_start, x_goal):
+        self.x_start = x_start
+        self.goal = x_goal
+
+        self.r = np.inf
+        self.V = set()
+        self.E = set()
+        self.QE = set()
+        self.QV = set()
+
+        self.V_old = set()
+
+
+class BITStar:
+    def __init__(self, x_start, x_goal, eta, iter_max):
+        self.x_start = Node(x_start[0], x_start[1])
+        self.x_goal = Node(x_goal[0], x_goal[1])
+        self.eta = eta
+        self.iter_max = iter_max
+
+        self.env = env.Env()
+        self.plotting = plotting.Plotting(x_start, x_goal)
+        self.utils = utils.Utils()
+
+        self.fig, self.ax = plt.subplots()
+
+        self.delta = self.utils.delta
+        self.x_range = self.env.x_range
+        self.y_range = self.env.y_range
+
+        self.obs_circle = self.env.obs_circle
+        self.obs_rectangle = self.env.obs_rectangle
+        self.obs_boundary = self.env.obs_boundary
+
+        self.Tree = Tree(self.x_start, self.x_goal)
+        self.X_sample = set()
+        self.g_T = dict()
+        self.f_T = dict()
+
+    def init(self):
+        self.Tree.V.add(self.x_start)
+        self.X_sample.add(self.x_goal)
+        self.g_T[self.x_goal] = np.inf
+        self.f_T[self.x_goal] = 0.0
+        self.g_T[self.x_start] = 0.0
+        self.f_T[self.x_start] = self.f_estimated(self.x_start)
+
+        cMin, theta = self.calc_dist_and_angle(self.x_start, self.x_goal)
+        C = self.RotationToWorldFrame(self.x_start, self.x_goal, cMin)
+        xCenter = np.array([[(self.x_start.x + self.x_goal.x) / 2.0],
+                            [(self.x_start.y + self.x_goal.y) / 2.0], [0.0]])
+
+        return theta, cMin, xCenter, C
+
+    def planning(self):
+        eTheta, cMin, xCenter, C = self.init()
+
+        for k in range(self.iter_max):
+            if not self.Tree.QE and not self.Tree.QV:
+                self.Prune(self.g_T[self.x_goal])
+                m = 200
+                self.X_sample.update(self.Sample(m, self.g_T[self.x_goal], cMin, xCenter, C))
+                self.Tree.V_old = copy.deepcopy(self.Tree.V)
+                self.Tree.QV = copy.deepcopy(self.Tree.V)
+                self.Tree.r = self.radius(len(self.Tree.V) + len(self.X_sample))
+
+            while self.BestVertexQueueValue() <= self.BestEdgeQueueValue():
+                self.ExpandVertex(self.BestInVertexQueue())
+
+            vm, xm = self.BestInEdgeQueue()
+            self.Tree.QE.remove((vm, xm))
+
+            if self.g_T[vm] + self.calc_dist(vm, xm) + self.h_estimated(xm) < self.g_T[self.x_goal]:
+                if self.g_estimated(vm) + self.cost(vm, xm) + self.h_estimated(xm) < self.g_T[self.x_goal]:
+                    if self.g_T[vm] + self.cost(vm, xm) < self.g_T[xm]:
+                        if xm in self.Tree.V:
+                            # remove edges
+                            for vl, vr in self.Tree.E:
+                                if vl == xm or vr == xm:
+                                    self.Tree.E.remove((vl, vr))
+                        else:
+                            self.X_sample.remove(xm)
+                            self.Tree.V.add(xm)
+                            self.Tree.QV.add(xm)
+
+                        self.Tree.E.add((vm, xm))
+
+    def ExpandVertex(self, v):
+        self.Tree.QV.remove(v)
+        X_near = {x for x in self.X_sample if self.calc_dist(x, v) <= self.Tree.r}
+
+        for x in X_near:
+            if self.g_estimated(v) + self.calc_dist(v, x) + self.h_estimated(x) < self.g_T[self.x_goal]:
+                self.Tree.QE.add((v, x))
+
+        if v not in self.Tree.V_old:
+            V_near = {w for w in self.Tree.V if self.calc_dist(w, v) <= self.Tree.r}
+
+            for w in V_near:
+                if (v, w) not in self.Tree.E and (w, v) not in self.Tree.E and \
+                        self.g_estimated(v) + self.calc_dist(v, w) + self.h_estimated(w) < self.g_T[self.x_goal] and \
+                        self.g_T[v] + self.calc_dist(v, w) < self.g_T[w]:
+                    self.Tree.QE.add((v, w))
+
+    def BestVertexQueueValue(self):
+        if not self.Tree.QV:
+            return np.inf
+
+        return min(self.g_T[v] + self.h_estimated(v) for v in self.Tree.QV)
+
+    def BestEdgeQueueValue(self):
+        if not self.Tree.QE:
+            return np.inf
+
+        return min(self.g_T[el] + self.calc_dist(el, er) + self.h_estimated(er)
+                   for el, er in self.Tree.QE)
+
+    def BestInVertexQueue(self):
+        v_value = {v: self.g_T[v] + self.h_estimated(v) for v in self.Tree.QV}
+
+        return min(v_value, key=v_value.get)
+
+    def BestInEdgeQueue(self):
+        e_value = {(el, er): self.g_T[el] + self.calc_dist(el, er) + self.h_estimated(er)
+                   for el, er in self.Tree.QE}
+
+        return min(e_value, key=e_value.get)
+
+    def radius(self, q):
+        lambda_X = 0
+        sigma = math.pi ** 2
+
+        for x in self.Tree.V:
+            if self.f_estimated(x) <= self.g_T[self.x_goal]:
+                lambda_X += 1
+
+        return 2 * self.eta * 1.5 ** 0.5 * (lambda_X / sigma * math.log(q) / q) ** 0.5
+
+    def Sample(self, m, cMax, cMin, xCenter, C):
+        if cMax < np.inf:
+            Sample = self.SampleEllipsoid(m, cMax, cMin, xCenter, C)
+        else:
+            Sample = self.SampleFreeSpace(m)
+
+        return Sample
+
+    def SampleEllipsoid(self, m, cMax, cMin, xCenter, C):
+        r = [cMax / 2.0,
+             math.sqrt(cMax ** 2 - cMin ** 2) / 2.0,
+             math.sqrt(cMax ** 2 - cMin ** 2) / 2.0]
+        L = np.diag(r)
+
+        ind = 0
+        delta = self.delta
+        Sample = set()
+
+        while ind < m:
+            xBall = self.SampleUnitNBall()
+            x_rand = C @ L @ xBall + xCenter
+            node = Node(x_rand[0], x_rand[1])
+            not_in_obs = ~self.utils.is_inside_obs(node)
+            in_x_range = self.x_range[0] + delta <= node.x <= self.x_range[1] - delta
+            in_y_range = self.y_range[0] + delta <= node.y <= self.y_range[1] - delta
+
+            if not_in_obs and in_x_range and in_y_range:
+                Sample.add(node)
+                ind += 1
+
+        return Sample
+
+    def SampleFreeSpace(self, m):
+        delta = self.utils.delta
+        Sample = set()
+
+        ind = 0
+        while ind < m:
+            node = Node((random.uniform(self.x_range[0] + delta, self.x_range[1] - delta),
+                         random.uniform(self.y_range[0] + delta, self.y_range[1] - delta)))
+            if self.utils.is_inside_obs(node):
+                continue
+            else:
+                Sample.add(node)
+                ind += 1
+
+        return Sample
+
+    @staticmethod
+    def SampleUnitNBall():
+        theta, r = random.uniform(0.0, 2 * math.pi), random.random()
+        x = r * math.cos(theta)
+        y = r * math.sin(theta)
+
+        return np.array([[x], [y], [0.0]])
+
+    def Prune(self, c):
+        for x in self.X_sample:
+            if self.f_estimated(x) >= c:
+                self.X_sample.remove(x)
+
+        for v in self.Tree.V:
+            if self.f_estimated(v) > c:
+                self.Tree.V.remove(v)
+
+        for v, w in self.Tree.E:
+            if self.f_estimated(v) > c or self.f_estimated(w) > c:
+                self.Tree.E.remove((v, w))
+
+        for v in self.Tree.V:
+            if v.g_T == np.inf:
+                self.X_sample.add(v)
+
+        for v in self.Tree.V:
+            if v.g_T == np.inf:
+                self.Tree.V.remove(v)
+
+    def cost(self, start, end):
+        if self.utils.is_collision(start, end):
+            return np.inf
+
+        return self.calc_dist(start, end)
+
+    def f_estimated(self, node):
+        return self.g_estimated(node) + self.h_estimated(node)
+
+    def g_estimated(self, node):
+        return self.calc_dist(self.x_start, node)
+
+    def h_estimated(self, node):
+        return self.calc_dist(node, self.x_goal)
+
+    @staticmethod
+    def RotationToWorldFrame(x_start, x_goal, L):
+        a1 = np.array([[(x_start.x - x_start.x) / L],
+                       [(x_goal.y - x_start.y) / L], [0.0]])
+        e1 = np.array([[1.0], [0.0], [0.0]])
+        M = a1 @ e1.T
+        U, _, V_T = np.linalg.svd(M, True, True)
+        C = U @ np.diag([1.0, 1.0, np.linalg.det(U) * np.linalg.det(V_T.T)]) @ V_T
+
+        return C
+
+    @staticmethod
+    def calc_dist(start, end):
+        return math.hypot(start.x - end.x, start.y - end.y)
+
+    @staticmethod
+    def calc_dist_and_angle(node_start, node_end):
+        dx = node_end.x - node_start.x
+        dy = node_end.y - node_start.y
+        return math.hypot(dx, dy), math.atan2(dy, dx)
diff --git a/Sampling_based_Planning/rrt_2D/informed_rrt_star.py b/Sampling_based_Planning/rrt_2D/informed_rrt_star.py
index 20c244c..f96f798 100644
--- a/Sampling_based_Planning/rrt_2D/informed_rrt_star.py
+++ b/Sampling_based_Planning/rrt_2D/informed_rrt_star.py
@@ -123,11 +123,12 @@ class IRrtStar:
                  math.sqrt(c_max ** 2 - c_min ** 2) / 2.0,
                  math.sqrt(c_max ** 2 - c_min ** 2) / 2.0]
             L = np.diag(r)
-            x_ball = self.SampleUnitNBall()
 
             while True:
+                x_ball = self.SampleUnitNBall()
                 x_rand = C @ L @ x_ball + x_center
-                if self.x_range[0] + self.delta <= x_rand[0] <= self.x_range[1] + self.delta:
+                if self.x_range[0] + self.delta <= x_rand[0] <= self.x_range[1] - self.delta and \
+                        self.y_range[0] + self.delta <= x_rand[1] <= self.y_range[1] - self.delta:
                     break
             x_rand = Node((x_rand[0], x_rand[1]))
         else: