Death of the trustee. In the case where the last remaining trustee is death, by considering s18(2) Trustee Act 1925, the deceased trustee's personal representative may be considered as a new trustee. These may include executor in the case of a will or administrator in a case where there is no will or considered s36(1) Trustee Act 1925 where the surviving trustee s36(1)(b) or person nominated will appoint a new trustee s36(1)(a).

Trustee who wants to retire may also consider section 36(1) Trustee Act 1925, where he / she can request to be discharged and the person nominated to appoint a new trustee s36(1)(a) or remaining trustee may appoint new trustee as in s36(1)(b).

s39 Trustee Act 1925, indicates that a trustee can retire voluntarily with the conditions that there are two people / trustees remaining after the retirement. The deed must also be signed by the existing trustee, retiring trustee and person with power to appoint trustee.

Removal of Trustee 
The reasons to end the appointment of trustee may include the retirement of the trustee. For instance the retirement of the solicitor from the practise or The trustee decides to live abroad out from the United Kingdom for extended period of time or involved in criminal activity (s36 Trustee Act 1925) or 
The beneficiaries and other trustee are dissatisfied with the action and performance of the trustee.( s19 Trust of the land and Appointment of Trustee 1996) for beneficiaries or
The trustee lack of capacity to exercise the function of trustee, the corporate trustee in liquidation or dissolved ( s41 Trustee Act 1925) 
The appointment of further trustee might be included by the settlor in the express provision of the trust instrument. 
There are 2 ways in which a trustee could retire from the existing trust : 
-1st based on the steps provided under the express provision of the trust instrument 
-2nd based on the statutory power which is recommended as enable the vesting of trust property to new trustee. In this way, a sole trustee or group of trustees who decide to retire at the same time will not escape fiduciary responsibility without requisite number of replacement trustees appointed at the same time where trust property being properly vested. 

Under s36(6) Trustee Act, additional trustee will be appointed when necessary and it must be done in writing and falls under the following circumstances where the trustee die or wishes to retire.
The appointment of new trustee will be restricted to make sure that the total number of maximum trustee is four.
s36(6)(a) indicates that a person may be nominated to appoint a new trustees and if no one is available therefore the trustee present at that time may appoint the additional trustee as in s36(6)(b)
​There is no obligation to appoint additional trustee unless the statutory requirement or express provision on the instrument required to do so.

General views of Appointment of TrusteeAppointment of trustee is possible via : 
-the trust instrument 
-appointed by the court s41 Trustee Act 1941
-appointed by the nominated person/ person with the duty to appoint new trustee s36(6)(a)Trustee Act 1925
-appointed by the existing trustee as in s36(6)(b) Trustee Act 1925
-appointed by surviving trustee in case the trustee die as in s36(1) Trustee Act 1926
-appointed by beneficiaries as in s19 Trust of the land and Appointment of Trustee Act 1996 

Appointment of Trustee
The trust instrument contained the name of the trustee. 
In case of will, the name of the executors might also be included as trustee.If you are nominated to be a trustee, you are not forced to accept the title / post.
A potential trustee may refuse the appointment which could be done via deed or sometimes based on the disclaimer which is inferred from the circumstances as in the case of re Clout and Frewer's Contract [1924] 2 Ch 230.
It is impossible to disclaim, once the trustee have accepted the role. 
​

How A Court Appoint A New Trustee 
Court will appoint a trustee if it is expedient to do so.The new trustee might be a new trustee on its own rights or new trustee as substitute or additional to the existing line of the trustee. Substitution of the new trustee is always being considered in cases where the trustee unable to exercise the function of the trustee , bankrupt or if the corporate trustee has been dissolved or in  liquidation.
Court appoint trustee under s41 Trustee Act 1925.
This not to be taken into lightly. The whole procedure is expensive and the outcome might not be expected. 
It was applied in the case of Ledderstedt v Broers [1884] 9 App Cas 371
" the overriding consideration is ---whether the trusts are being properly excluded " as indicated by Lewison J in Thomas & Agnes Carvel Foundation v Carvel [2007] EWHC 1314
or the "the main guide must be the welfare of the beneficiaries" as indicated by Lord Blackburn.

Trust Property is Land 
Two trustees are needed when the land is a trust property so that a valid receipt of any sale of land could be achieved as in s27(2) Law of Property Act 1925 and s14(2) Trustee Act 1925.
However, the maximum number of trustee could never exceeded four as in s34(2) Trustee Act 1925.