diff --git a/docs/glossary.md b/docs/glossary.md
index 3317c9a27b..20cca9204f 100644
--- a/docs/glossary.md
+++ b/docs/glossary.md
@@ -30,26 +30,39 @@ Here are some definitions of some key ideas encountered in this documentation.
 tree
 : A "gene tree", i.e., the genealogical tree describing how a collection of
   genomes (usually at the tips of the tree) are related to each other at some
-  chromosomal location. See {ref}`sec_nodes_or_individuals` for discussion
-  of what a "genome" is.
+  chromosomal {ref}`position <sec_data_model_definitions_position>` or location.
+  As the trees may vary depending on this location, they are also known as "local
+  trees". See {ref}`sec_nodes_or_individuals` for discussion of what a "genome" is.
 
 (sec_data_model_definitions_tree_sequence)=
 
 tree sequence
-: A "succinct tree sequence" (or tree sequence, for brevity) is an efficient
-  encoding of a sequence of correlated trees, such as one encounters looking
-  at the gene trees along a genome. A tree sequence efficiently captures the
-  structure shared by adjacent trees, (essentially) storing only what differs
-  between them.
+: A "succinct tree sequence" (or tree sequence, for brevity) is an object
+  that stores the genetic ancestry and mutational history of a set of
+  aligned DNA sequences or genomes. The name reflects the idea that a common
+  way to treat genetic ancestry is as a sequence of correlated
+  {ref}`trees <sec_data_model_definitions_tree>` at different chromosomal
+  {ref}`positions <sec_data_model_definitions_position>`.
+  Branches that are shared between these trees are efficiently stored as a
+  single {ref}`edge <sec_data_model_definitions_edge>`, and adjacent trees
+  may differ by only a few such edges. These edges connect
+  {ref}`nodes <sec_data_model_definitions_node>` (genomes) in
+  the tree sequence, forming  a
+  network or graph. Graphs of this sort are sometimes called ancestral
+  recombination graphs (ARGs), hence tree sequences provide a
+  flexible way to encode multiple types of ARG.
 
 (sec_data_model_definitions_node)=
 
 node
-: Each branching point in each tree is associated with a particular genome
+: Any point in a tree can be associated with a particular genome
   in a particular ancestor, called a "node".  Since each node represents a
-  specific genome it has a unique `time`, thought of as its birth time,
-  which determines the height of any branching points it is associated with.
-  See {ref}`sec_nodes_or_individuals` for discussion of what a "node" is.
+  specific genome it has a unique `time`, thought of as its birth time. Nodes
+  may or may not correspond to branching points, either in a local
+  {ref}`tree <sec_data_model_definitions_tree>` or in the whole graph.
+  However a branching point must always be associated with a node.
+  See {ref}`sec_nodes_or_individuals` for discussion of what a "node"
+  represents.
 
 (sec_data_model_definitions_individual)=
 
@@ -66,7 +79,7 @@ individual
 sample
 : The focal nodes of a tree sequence, usually thought of as those from which
   we have obtained data. The specification of these affects various
-  methods: (1) {meth}`TreeSequence.variants` and
+  methods: {meth}`TreeSequence.variants` and
   {meth}`TreeSequence.haplotypes` will output the genotypes of the samples,
   and {attr}`Tree.roots` only return roots ancestral to at least one
   sample.
@@ -81,13 +94,15 @@ edge
 : The topology of a tree sequence is defined by a set of **edges**. Each
   edge is a tuple `(left, right, parent, child)`, which records a
   parent-child relationship among a pair of nodes on the
-  on the half-open interval of chromosome `[left, right)`.
+  on the half-open interval `[left, right)` along the genome. The difference
+  between `left` and `right` is known as the "span" of the edge.
 
 (sec_data_model_definitions_site)=
 
 site
 : Tree sequences can define the mutational state of nodes as well as their
-  topological relationships. A **site** is thought of as some position along
+  topological relationships. A **site** is thought of as some
+  {ref}`position <sec_data_model_definitions_position>` along
   the genome at which variation occurs. Each site is associated with
   a unique position and ancestral state.
 
@@ -114,6 +129,14 @@ migration
 population
 : A grouping of nodes, e.g., by sampling location.
 
+(sec_data_model_definitions_position)=
+
+position
+: A location along the genome, from 0 to the 
+  {ref}`sequence length<sec_data_model_definitions_sequence_length>`. In `tskit`
+  positions are stored as floating-point numbers, although it is common to
+  restrict positions to occur at discrete integer locations.
+
 (sec_data_model_definitions_provenance)=
 
 provenance