diff --git a/networkx/generators/random_graphs.py b/networkx/generators/random_graphs.py
index e4f2c569d..745f64e4d 100644
--- a/networkx/generators/random_graphs.py
+++ b/networkx/generators/random_graphs.py
@@ -1000,6 +1000,11 @@ def random_lobster(n, p1, p2, seed=None):
     leaf nodes. A caterpillar is a tree that reduces to a path graph
     when pruning all leaf nodes; setting `p2` to zero produces a caterpillar.
 
+    This implementation iterates on the probabilities `p1` and `p2` to add
+    edges at levels 1 and 2, respectively. Graphs are therefore constructed
+    iteratively with uniform randomness at each level rather than being selected
+    uniformly at random from the set of all possible lobsters.
+
     Parameters
     ----------
     n : int
@@ -1011,19 +1016,29 @@ def random_lobster(n, p1, p2, seed=None):
     seed : integer, random_state, or None (default)
         Indicator of random number generation state.
         See :ref:`Randomness<randomness>`.
+
+    Raises
+    ------
+    NetworkXError
+        If `p1` or `p2` parameters are >= 1 because the while loops would never finish.
     """
+    p1, p2 = abs(p1), abs(p2)
+    if any([p >= 1 for p in [p1, p2]]):
+        raise nx.NetworkXError("Probability values for `p1` and `p2` must both be < 1.")
+
     # a necessary ingredient in any self-respecting graph library
     llen = int(2 * seed.random() * n + 0.5)
     L = path_graph(llen)
     # build caterpillar: add edges to path graph with probability p1
     current_node = llen - 1
     for n in range(llen):
-        if seed.random() < p1:  # add fuzzy caterpillar parts
+        while seed.random() < p1:  # add fuzzy caterpillar parts
             current_node += 1
             L.add_edge(n, current_node)
-            if seed.random() < p2:  # add crunchy lobster bits
+            cat_node = current_node
+            while seed.random() < p2:  # add crunchy lobster bits
                 current_node += 1
-                L.add_edge(current_node - 1, current_node)
+                L.add_edge(cat_node, current_node)
     return L  # voila, un lobster!
 
 

diff --git a/networkx/generators/tests/test_random_graphs.py b/networkx/generators/tests/test_random_graphs.py
index f958dceab..8f2d68415 100644
--- a/networkx/generators/tests/test_random_graphs.py
+++ b/networkx/generators/tests/test_random_graphs.py
@@ -91,7 +91,38 @@ class TestGeneratorsRandom:
         constructor = [(10, 20, 0.8), (20, 40, 0.8)]
         G = random_shell_graph(constructor, seed)
 
+        def is_caterpillar(g):
+            """
+            A tree is a caterpillar iff all nodes of degree >=3 are surrounded
+            by at most two nodes of degree two or greater.
+            ref: http://mathworld.wolfram.com/CaterpillarGraph.html
+            """
+            deg_over_3 = [n for n in g if g.degree(n) >= 3]
+            for n in deg_over_3:
+                nbh_deg_over_2 = [nbh for nbh in g.neighbors(n) if g.degree(nbh) >= 2]
+                if not len(nbh_deg_over_2) <= 2:
+                    return False
+            return True
+
+        def is_lobster(g):
+            """
+            A tree is a lobster if it has the property that the removal of leaf
+            nodes leaves a caterpillar graph (Gallian 2007)
+            ref: http://mathworld.wolfram.com/LobsterGraph.html
+            """
+            non_leafs = [n for n in g if g.degree(n) > 1]
+            return is_caterpillar(g.subgraph(non_leafs))
+
         G = random_lobster(10, 0.1, 0.5, seed)
+        assert max([G.degree(n) for n in G.nodes()]) > 3
+        assert is_lobster(G)
+        pytest.raises(NetworkXError, random_lobster, 10, 0.1, 1, seed)
+        pytest.raises(NetworkXError, random_lobster, 10, 1, 1, seed)
+        pytest.raises(NetworkXError, random_lobster, 10, 1, 0.5, seed)
+
+        # docstring says this should be a caterpillar
+        G = random_lobster(10, 0.1, 0.0, seed)
+        assert is_caterpillar(G)
 
         # difficult to find seed that requires few tries
         seq = random_powerlaw_tree_sequence(10, 3, seed=14, tries=1)

Wrong random_lobster implementation?
Hi, it seems that [networkx.random_lobster's implementation logic](https://github.com/networkx/networkx/blob/4e9771f04192e94a5cbdd71249a983d124a56593/networkx/generators/random_graphs.py#L1009) is not aligned with the common definition as given in [wolfram mathworld](http://mathworld.wolfram.com/LobsterGraph.html) or [Wikipedia](https://en.wikipedia.org/wiki/List_of_graphs#Lobster).
For example, it can not produce the simplest lobster graph examples such as ![lobster_example](https://www.researchgate.net/profile/Dr_Nasreen_Khan/publication/256292297/figure/fig5/AS:669566174756876@1536648424468/A-diameter-2-lobster-graph.png).
networkx/generators/tests/test_random_graphs.py::TestGeneratorsRandom::test_random_graph
networkx/generators/tests/test_random_graphs.py::TestGeneratorsRandom::test_dual_barabasi_albert networkx/generators/tests/test_random_graphs.py::TestGeneratorsRandom::test_extended_barabasi_albert networkx/generators/tests/test_random_graphs.py::TestGeneratorsRandom::test_random_zero_regular_graph networkx/generators/tests/test_random_graphs.py::TestGeneratorsRandom::test_gnp networkx/generators/tests/test_random_graphs.py::TestGeneratorsRandom::test_gnm networkx/generators/tests/test_random_graphs.py::TestGeneratorsRandom::test_watts_strogatz_big_k networkx/generators/tests/test_random_graphs.py::TestGeneratorsRandom::test_random_kernel_graph