Balfour, Sir Isaac Bayley (1853–1922), botanist, was born at 27 Inverleith Row, Edinburgh, on 31 March 1853, the third child and second son of John Hutton Balfour (1808–1884), professor of botany in the University of Edinburgh, and his wife, Marion, daughter of Isaac Bayley, of Edinburgh, an attorney. As though predestined to repeat his father's career, Balfour spent much time in boyhood roaming the Royal Botanic Garden, where he won an early grounding in plantsmanship at the hands of the curator and staff. His formal education was at Edinburgh Academy and Edinburgh University, where he specialized in botany, graduating BSc in 1873. He later studied under Julius von Sachs at Würzburg and with Anton de Bary at Strasbourg. From 1875 to 1878 he acted as lecturer in botany to the Royal Veterinary College; for this he was required to obtain a medical qualification and so became assistant to Charles Wyville Thomson, professor of natural history at Edinburgh, for two years, working with T. H. Huxley, Thomson's substitute during his absence on the Challenger voyage. He graduated MB in 1877 and MD in 1883.

In 1874 Balfour had served as botanist-cum-geologist on an expedition sent to Rodriguez Island in the Indian Ocean to observe the transit of Venus. His report on the island's flora, published in the Royal Society's Philosophical Transactions in 1879, established him as a plant taxonomist and gained him a DSc. That same year Balfour spent seven weeks on another Indian Ocean island, Socotra, investigating its flora and geology. Working up the rich collections he made there occupied his spare time over the next eight years and his Botany of Socotra (1888), including descriptions of some three hundred new species, was to be his most substantial published work.

Balfour crowned that golden year of 1879 by obtaining the regius chair of botany at Glasgow. However, his stay in that post proved short, for in 1884 he secured the Sherardian chair of botany at Oxford, combined with a fellowship of Magdalen College. In that year he married Agnes Boyd, daughter of Robert Balloch, merchant, of Glasgow. They had one son and one daughter.

During his tenure at Oxford Balfour established relations with the Clarendon Press, inducing it to found, in 1887, the Annals of Botany, a quarterly journal intended to publish results from experimental botanical research. Under his editorship, the press also produced a series of translations of continental textbooks. In 1888 Balfour returned to his native Edinburgh on his appointment as professor of botany in the university, queen's botanist in Scotland, and regius keeper of the Royal Botanic Garden, three positions earlier held by his father and which he was to hold for a third of a century, until shortly before his death.

Balfour's main interest was in rhododendrons and primulas, on which he was recognized as the foremost authority; he also did valuable work on the propagation of plants and the germination of seeds. But his influence was greatest as an administrator, in which role he displayed exceptional ability. In each of the universities which he served he found a department needing reorganization: in Glasgow he saved the herbarium and secured rebuilding of the plant houses, in his few years at Oxford, where he found botany in a sorry state, he did the same, and in both he reorganized the teaching on modern lines. But he did most at Edinburgh: there he reconstituted the botanical garden, rebuilt its plant houses, enlarged the laboratories, and created a new rock-garden, no less renowned than his father's. For a quarter of a century Balfour was the most effective all-round botanist in Britain.

Balfour was elected a fellow of the Royal Society in 1884 and created KBE in 1920. The University of Edinburgh conferred on him an honorary LLD in 1921. The next year ill health prevented his assuming the presidency of the British Association and realizing his intention of writing a history of the Edinburgh botanic garden, and his death followed, at Court Hill, Haslemere, on 30 November 1922.