A Tale of Three Students
1) What legal or equitable interests may exist here?
Denise is described as the owner of the house, so she presumably has a legal estate in the land. She probably has a freehold estate, although it is possible that she has a long leasehold (see Section 2.3.1). Freeholds are considered in more detail in Chapter 4, leaseholds in Chapters 5 and 6.
The status of the three students is less clear. They might be tenants. If so, they will have a leasehold estate. If so, their lease (or leases) will bind any superior owner of the land, so it will be binding on anyone who buys the house from Denise (see Section 2.3.2). However, it is also possible that the three students do not have a lease. The requirements for a lease are set out in Section 5.4. If the students do not have a lease, then they probably have some sort of contractual licence (see Section 3.5). Contractual licences will not normally bind a purchaser of an estate in the land.
Interests in the land
Brian’s dad is proposing to raise part of the purchase price by mortgage. A mortgage is an interest secured on the land: see LPA 1925, s 1(2)(c) (mortgages are considered in Chapter 7).
We don’t know what rights, if any, the neighbours had to use the shared path. However, it is possible that this was a right of way capable of being an easement: see LPA 1925, s1(2)(a) (easements are considered in more detail in Chapter 8). There may also have been a promise not to build on the garden amounting to a restrictive covenant. A freehold covenant may be enforceable against a subsequent owner of the land in equity (see Chapter 9).
Although not strictly relevant to this question, it is also helpful to consider what Denise actually owns. Whether or not the furniture (including the washing machine) belongs to Denise or her former tenants will depend upon whether the objects are fixtures or chattels (see Section 2.2.1).
2) If the interest is a beneficial interest under a trust, has it been overreached?
There is no evidence of a beneficial interest under a trust, so, for the moment, at least, the doctrine of overreaching (see Section 11.6.1 for more detail) is not relevant. However, if the house was sold to Alexia, Brian’s father and Brian, the result would be a trust, because whenever two or more people own land concurrently they must hold the land as trustees on trust for themselves (see Chapter 10). Even if Brian’s name was not ultimately on the title deeds, he would still have beneficial interest if he contributed his savings towards the purchase of the house (see Bull v Bull  1 QB 234 (CA) in Sections 18.104.22.168 and 2.4 and Chapter 14)
3) What else do we need to know to determine whether the interests are binding on the owner of the land?
Once we have identified the various possible interests in the land, we need to know whether Denise’s title is already registered or whether it has yet to be registered. The detailed rules for determining which, if any, of the interests will be binding on a purchaser of the house will depend upon which set of rules are being used. The rules of registered title are considered in Chapter 15 and the rules of unregistered title in Chapter 16.